#7143749

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2262 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)(17,20)(21,31)(22,27)(23,32)(26,33)(28,30)(34,38)(35,42)(36,37)(40,41)(43,44);(6,9)(7,11)(13,14,20,32,18,23,17,19)(15,21,33,42,25,35,26,31)(16,27,30,37,29,36,28,22)(24,38,41,43,39,44,40,34);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,25)(14,21,23,35)(16,24,29,39)(17,26,20,33)(19,31,32,42)(22,34,37,43)(27,38,36,44)(28,40,30,41);(6,9)(10,12)(13,16,18,29)(14,22,23,37)(15,24,25,39)(17,30,20,28)(19,27,32,36)(21,34,35,43)(26,41,33,40)(31,38,42,44);(5,7,8,11)(6,10,9,12)(15,25)(21,35)(24,39)(26,33)(31,42)(34,43)(38,44)(40,41)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7050945);