#484202

order	512
nilpotency class	4
isoclinism	isoclinic class of #484199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60770	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1081 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1081 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(5,8)(9,10,15,20)(11,21,25,49)(12,17,26,40)(13,29,33,47)(14,34,31,19)(16,36,39,23)(18,43,45,32)(22,48,50,70)(24,53,35,44)(27,56,54,69)(28,59,55,42)(30,46,61,38)(37,62,63,51)(41,66,65,58)(52,71,60,67)(57,68,72,64);(1,2)(3,5)(4,6)(7,8)(9,11,15,25)(10,16,20,39)(12,22,26,50)(13,30,33,61)(14,35,31,24)(17,37,40,63)(18,44,45,53)(19,46,34,38)(21,47,49,29)(23,43,36,32)(27,57,54,72)(28,60,55,52)(41,67,65,71)(42,68,59,64)(48,69,70,56)(51,66,62,58);(1,3)(2,5)(4,7)(6,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,26)(16,37)(18,41)(19,42)(20,40)(21,48)(23,51)(24,52)(25,50)(29,56)(30,57)(31,55)(32,58)(33,54)(34,59)(35,60)(36,62)(38,64)(39,63)(43,66)(44,67)(45,65)(46,68)(47,69)(49,70)(53,71)(61,72);(12,26)(17,40)(22,50)(27,54)(28,55)(37,63)(41,65)(42,59)(48,70)(51,62)(52,60)(56,69)(57,72)(58,66)(64,68)(67,71)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=gi,g^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391398);