#379011

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=f,d^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,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, 286207);