#513024

order	512
nilpotency class	5
isoclinism	isoclinic class of #513022
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 ◁ C₂×C₄×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 345 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:

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

Polycyclic presentation:

2^9|b^2=i,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,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, #72225113, #72225114, …
maximal normal subgroup of	#72224992, #72225113, #72225114, …
maximal quotient of	C₂×#513024

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420220);