#373462

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

863 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=f,d^2=fg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,e^b=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280658);