#412389

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1460 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)(6,8)(9,10)(11,20)(12,24)(13,28)(14,18)(15,19)(16,22)(17,26)(21,37)(23,33)(25,31)(27,35)(29,32)(30,36)(34,38)(39,40);(9,11,13,22,14,23,15,29)(10,16,19,20,18,32,28,33)(12,25,27,21,26,39,30,38)(17,34,35,40,24,37,36,31);(2,4)(9,12)(10,17)(11,21)(13,27)(14,26)(15,30)(16,31)(18,24)(19,35)(20,34)(22,39)(23,38)(25,29)(28,36)(32,40)(33,37);(5,6,7,8)(10,18)(12,26)(16,32)(19,28)(20,33)(21,38)(25,39)(27,30)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=di,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, 319585);