#7149362

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
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1875 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)(14,19)(15,23)(17,31)(20,36)(21,34)(22,35)(24,33)(25,28)(27,40)(29,32)(38,44)(39,41);(6,9)(7,11)(13,14,17,19,18,22,31,35)(15,24,26,36,23,20,37,33)(16,27,29,40,30,34,32,21)(25,41,42,38,28,44,43,39);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,28)(17,26)(18,23)(19,33)(21,39)(22,24)(25,30)(27,41)(29,43)(31,37)(32,42)(34,44)(35,36)(38,40);(6,9)(10,12)(13,16,18,30)(14,21,22,40)(15,25,23,28)(17,32,31,29)(19,34,35,27)(20,38,24,39)(26,43,37,42)(33,41,36,44);(5,7,8,11)(6,10,9,12)(14,22)(15,23)(19,35)(21,40)(25,28)(26,37)(27,34)(42,43)

Polycyclic presentation:

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

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, 7056558);