#7146010

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2189 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7053206);