#351485

order	512
nilpotency class	3
isoclinism	isoclinic class of #348438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=efhi,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ci,g^d=ghi

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, 258681);