#495563

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60682	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

812 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)(3,6)(4,5)(7,8)(9,10,14,20,16,21,15,36)(11,22,25,47,27,48,26,55)(12,18,30,44,28,42,31,60)(13,32,33,38,35,61,34,19)(17,37,40,54,41,53,46,24)(23,45,51,68,49,66,52,71)(29,57,58,63,56,72,59,43)(39,62,65,69,64,70,67,50);(9,11,13,24,14,25,33,37,16,27,35,54,15,26,34,53)(10,17,38,48,36,46,32,47,21,41,19,22,20,40,61,55)(12,23,29,50,30,51,58,62,28,49,56,69,31,52,59,70)(18,39,63,66,60,67,57,68,42,64,43,45,44,65,72,71);(2,5)(6,8)(9,12)(10,18)(11,23)(13,29)(14,30)(15,31)(16,28)(17,39)(19,43)(20,44)(21,42)(22,45)(24,50)(25,51)(26,52)(27,49)(32,57)(33,58)(34,59)(35,56)(36,60)(37,62)(38,63)(40,65)(41,64)(46,67)(47,68)(48,66)(53,70)(54,69)(55,71)(61,72);(1,3,4,7)(2,6,5,8)(12,28)(18,42)(23,49)(29,56)(30,31)(39,64)(43,63)(44,60)(45,66)(50,69)(51,52)(57,72)(58,59)(62,70)(65,67)(68,71)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=f,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402759);