#362235

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,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, 269431);