#473935

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egh,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381131);