#406243

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1718 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)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,28,36)(16,21,31,38)(17,23,18,22)(20,29,35,27)(26,37,42,41)(30,34,43,44)(32,40,33,39);(2,4)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,30)(17,29)(18,27)(19,28)(21,37)(22,36)(23,25)(24,35)(26,33)(31,43)(32,42)(34,40)(38,41)(39,44);(13,16)(14,21)(15,26,28,42)(17,32)(18,33)(19,31)(20,34,35,44)(22,39)(23,40)(24,38)(25,41,36,37)(27,43,29,30);(6,10)(9,12)(16,31)(21,38)(26,42)(30,43)(32,33)(34,44)(37,41)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di

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, 313439);