#7146135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei

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, 7053331);