#423661

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1605 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,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, 330857);