#420471

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1771 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=e,d^2=e,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,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327667);