#408257

order	512
nilpotency class	3
isoclinism	isoclinic class of #393899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1060 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)(3,5)(4,6)(7,8)(9,10,14,19,16,20,15,32)(11,21,24,35,25,37,26,17)(12,18,28,13,29,30,33,31)(22,36,27,34,39,38,40,23);(1,3,4,7)(2,5,6,8)(9,11,15,24,16,25,14,26)(10,17,32,21,20,35,19,37)(12,27,33,39,29,40,28,22)(13,23,18,36,31,34,30,38);(9,12)(10,18)(11,22)(13,19)(14,28)(15,33)(16,29)(17,34)(20,30)(21,38)(23,35)(24,27)(25,39)(26,40)(31,32)(36,37);(2,6)(5,8)(9,13,14,30,16,31,15,18)(10,12,19,28,20,29,32,33)(11,23,26,38,25,34,24,36)(17,27,37,22,35,40,21,39)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315453);