#7143709

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

2230 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)(17,29)(20,31)(21,32)(22,33)(25,40)(26,36)(27,30)(34,42)(35,43)(37,44)(39,41);(6,9)(7,11)(13,14,17,19,18,22,29,33)(15,20,25,31,24,35,40,43)(16,26,27,36,28,32,30,21)(23,37,39,44,38,42,41,34);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,20,22,35)(16,23,28,38)(17,25,29,40)(19,31,33,43)(21,34,36,44)(26,37,32,42)(27,39,30,41);(6,9)(10,12)(13,16,18,28)(14,21,22,36)(15,23,24,38)(17,30,29,27)(19,32,33,26)(20,34,35,44)(25,41,40,39)(31,42,43,37);(5,7,8,11)(6,10,9,12)(15,24)(20,35)(23,38)(25,40)(31,43)(34,44)(37,42)(39,41)

Polycyclic presentation:

2^9|b^2=f,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=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, 7050905);