#381330

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(9,10)(11,15,20,29)(12,22,23,37)(13,18)(14,17)(16,24,31,38)(19,34)(21,28,26,32)(25,33)(27,39)(30,36)(35,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,20)(14,26)(16,30)(17,32)(18,29)(21,25)(22,34)(23,35)(24,39)(27,31)(28,33)(36,38)(37,40);(2,5)(6,8)(9,12,13,23)(10,16)(11,19,20,35)(14,24,25,38)(15,27)(17,22)(18,31)(21,36,26,39)(28,34)(29,30)(32,40)(33,37);(1,3,4,7)(2,6,5,8)(12,23)(14,25)(17,33)(19,35)(21,26)(22,37)(28,32)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288526);