#7149402

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#30446	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1904 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)(15,24)(17,20)(21,36)(22,28)(23,34)(25,33)(26,29)(30,32)(37,41)(38,44)(39,40);(6,9)(7,11)(13,14,20,34,18,23,17,19)(15,25,35,33,24,21,27,36)(16,28,32,39,31,40,30,22)(26,41,43,37,29,44,42,38);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,29)(17,27)(18,24)(19,33)(20,35)(22,38)(23,25)(26,31)(28,41)(30,43)(32,42)(34,36)(37,39)(40,44);(6,9)(10,12)(13,16,18,31)(14,22,23,39)(15,26,24,29)(17,32,20,30)(19,28,34,40)(21,37,25,38)(27,43,35,42)(33,44,36,41);(5,7,8,11)(6,10,9,12)(14,23)(15,24)(19,34)(22,39)(26,29)(27,35)(28,40)(42,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,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=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056598);