#408198

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1027 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)(3,8)(4,6)(5,7)(10,16)(12,26)(13,14)(17,30)(18,33)(19,20)(21,32)(22,36)(23,24)(27,28)(35,39)(37,38);(1,3,5,4)(2,6,7,8)(9,10)(11,21)(12,18)(13,20)(14,16)(15,19)(17,23)(22,35)(24,32)(25,30)(26,29)(27,34)(28,33)(31,37)(36,40)(38,39);(9,11)(10,17)(12,22)(13,23)(14,24)(15,25)(16,21)(18,31)(19,32)(20,30)(26,36)(27,37)(28,38)(29,35)(33,40)(34,39);(1,4,5,3)(2,6,7,8)(9,12,13,27,15,28,14,26)(10,18,20,34,19,33,16,29)(11,22,23,37,25,38,24,36)(17,31,30,39,32,40,21,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315394);