#415861

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

1610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,22)(15,19,24,33)(16,20,27,36)(17,29,31,21)(23,32,39,44)(25,41,40,34)(26,43,42,35)(28,38,30,37);(6,9)(10,11)(13,15)(14,19)(16,26,27,42)(17,25)(18,24)(20,35,36,43)(21,34)(22,33)(23,28,39,30)(29,41)(31,40)(32,37,44,38);(2,4)(13,16)(14,20)(15,23)(17,30)(18,27)(19,32)(21,38)(22,36)(24,39)(25,42)(26,40)(28,31)(29,37)(33,44)(34,43)(35,41);(5,7,8,12)(6,10,9,11)(15,24)(19,33)(23,39)(25,40)(26,42)(32,44)(34,41)(35,43)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=di,e^c=ei,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, 323057);