#362233

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

994 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269429);