#513025

order	512
nilpotency class	5
isoclinism	isoclinic class of #513025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#33669
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#13667	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(3,13)(4,18)(5,22)(6,25)(8,27)(10,33)(11,35)(12,32)(14,40)(15,41)(16,45)(17,42)(19,46)(20,24)(21,47)(23,55)(26,54)(28,57)(29,39)(30,60)(31,51)(34,37)(36,64)(38,58)(43,53)(44,56)(48,49)(50,61)(52,62)(59,63);(1,2)(3,8)(4,19)(5,11)(6,12)(7,9)(10,21)(13,39)(14,38)(15,29)(16,30)(17,28)(18,46)(20,50)(22,35)(23,52)(24,37)(25,32)(26,31)(27,41)(33,47)(34,53)(36,48)(40,57)(42,58)(43,61)(44,59)(45,63)(49,64)(51,54)(55,62)(56,60);(1,3)(2,8)(4,20)(5,24)(6,14)(7,15)(9,29)(10,34)(11,37)(12,38)(13,18)(16,22)(17,25)(19,50)(21,53)(23,56)(26,43)(27,33)(28,32)(30,35)(31,61)(36,63)(39,46)(40,55)(41,47)(42,49)(44,54)(45,48)(51,59)(52,60)(57,62)(58,64);(2,9)(3,14,15,17)(4,21)(8,28,29,38)(11,31)(12,32)(13,40,41,42)(16,43,44,24)(18,47)(20,45,53,56)(23,48)(27,57,39,58)(30,37,59,61)(34,63,50,60)(35,51)(49,55)

Polycyclic presentation:

2^9|d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^b=di,f^b=fi,d^c=dh,e^c=ei,f^c=fg,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224992, C₂×#513025, #72320907, …
maximal normal subgroup of	#72224992, C₂×#513025, #72320907, …
maximal quotient of	C₂×#513025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420221);