#409525

order	512
nilpotency class	3
isoclinism	isoclinic class of #402446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

711 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:

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

Polycyclic presentation:

2^9|a^2=egi,b^2=h,c^2=ef,d^2=e,e^2=i,f^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,f^b=fi,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, 316721);