#481476

order	512
nilpotency class	4
isoclinism	isoclinic class of #481466
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60459	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1149 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388672);