#501021

order	512
nilpotency class	4
isoclinism	isoclinic class of #501020
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24354
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#62636	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

981 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 981 conjugacy classes of subgroups — too many to draw. Only its 160 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)(3,13,17,8)(4,18)(5,10)(6,11)(7,12)(9,20)(14,43,45,30)(15,44,48,31)(16,41,28,32)(19,50,52,33)(21,51)(22,42)(23,36)(24,37)(25,58,55,38)(26,59)(27,40)(29,35)(34,53)(39,56)(46,62,49,60)(47,63,54,61)(57,64);(1,3)(2,8)(4,19)(5,17)(6,25)(7,28)(9,33)(10,13)(11,38)(12,41)(14,21)(15,27)(16,24)(18,50)(20,52)(22,54)(23,55)(26,49)(29,47)(30,34)(31,40)(32,37)(35,63)(36,58)(39,62)(42,61)(43,51)(44,64)(45,53)(46,56)(48,57)(59,60);(1,4)(2,9)(3,14)(5,20)(6,26)(7,29)(8,30)(10,18)(11,39)(12,42)(13,43)(15,47)(16,49)(17,45)(19,25)(21,27)(22,24)(23,56)(28,46)(31,61)(32,62)(33,38)(34,40)(35,37)(36,59)(41,60)(44,63)(48,54)(50,58)(51,64)(52,55)(53,57);(1,3)(2,8)(4,14)(5,17)(6,15)(7,16)(9,30)(10,13)(11,31)(12,32)(18,43)(19,21)(20,45)(22,46)(23,48)(24,28)(25,27)(26,47)(29,49)(33,34)(35,60)(36,44)(37,41)(38,40)(39,61)(42,62)(50,51)(52,53)(54,56)(55,57)(58,64)(59,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319971, #72343955, #72343960
maximal normal subgroup of	#72319971, #72343955, #72343960

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408217);