#379010

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
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)(9,10,14,19,15,20,29,38)(11,16,13,18,22,31,28,37)(12,17,26,35,27,36,25,34)(21,30,24,33,39,40,23,32);(1,2,4,3)(9,11,14,13,15,22,29,28)(10,16,19,18,20,31,38,37)(12,23,25,39,27,24,26,21)(17,32,34,40,36,33,35,30);(5,6,7,8)(9,12)(10,17)(11,21)(13,24)(14,26)(15,27)(16,30)(18,33)(19,35)(20,36)(22,39)(23,28)(25,29)(31,40)(32,37)(34,38);(1,3,4,2)(9,13,14,22,15,28,29,11)(10,18,19,31,20,37,38,16)(12,24,26,39,27,23,25,21)(17,33,35,40,36,32,34,30)

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=de,c^b=cg,d^b=di,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, 286206);