#394947

order	512
nilpotency class	3
isoclinism	isoclinic class of #394113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23393	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1543 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,24)(13,18)(14,27)(15,19)(17,26)(20,29)(21,36)(22,31)(23,32)(25,34)(28,33)(30,35)(37,39)(38,40);(2,5)(6,8)(9,11)(10,16)(12,21)(13,20)(14,22)(15,23)(17,30)(18,29)(19,32)(24,36)(25,37)(26,35)(27,31)(28,38)(33,40)(34,39);(11,20)(13,14)(16,29)(18,27)(21,35)(22,23)(24,34)(26,28)(30,39)(31,32)(36,40)(37,38);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,26)(14,28)(15,25)(16,30)(18,34)(19,33)(20,35)(22,38)(23,37)(24,27)(29,39)(31,36)(32,40)

Polycyclic presentation:

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

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, 302143);