#408549

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

630 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)(3,5)(4,6)(7,8)(9,10,15,20)(11,21)(12,27,30,17)(13,18,31,29)(14,19,22,33)(16,32)(23,24)(25,36)(26,37)(28,40)(34,35)(38,39);(1,3,4,7)(2,5,6,8)(9,11,22,25,15,26,14,32)(10,16,33,21,20,36,19,37)(12,28,18,35,30,23,29,38)(13,24,27,39,31,40,17,34);(9,12,14,29,15,30,22,18)(10,17,19,31,20,27,33,13)(11,23,25,38,26,28,32,35)(16,34,21,24,36,39,37,40);(2,6)(5,8)(9,13,22,27,15,31,14,17)(10,18,33,30,20,29,19,12)(11,24,25,39,26,40,32,34)(16,35,21,23,36,38,37,28)

Polycyclic presentation:

2^9|a^2=f,b^2=efh,c^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,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, 315745);