#393596

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

846 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,e^b=ei,d^c=di

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, 300792);