#7143210

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2017 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)(13,14,20,25)(15,26,30,41)(16,22,17,32)(18,24,19,23)(21,35,38,29)(27,42,33,37)(28,36,44,43)(31,40,34,39);(6,9)(7,11)(13,15,20,30)(14,21,25,38)(16,27,17,33)(18,29,19,35)(22,36,32,43)(23,26,24,41)(28,31,44,34)(37,40,42,39);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,17)(14,22,25,32)(15,27,30,33)(18,34,19,31)(21,36,38,43)(23,40,24,39)(26,42,41,37)(28,35,44,29);(6,9)(10,12)(13,17,20,16)(14,22,25,32)(15,28,30,44)(18,34,19,31)(21,37,38,42)(23,39,24,40)(26,43,41,36)(27,35,33,29);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,25,24)(15,29,30,35)(16,31,17,34)(21,26,38,41)(22,39,32,40)(27,44,33,28)(36,37,43,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7050406);