#7143789

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

2233 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)(13,14,20,17)(15,24,27,37)(16,22,30,32)(18,23,19,33)(21,34,31,26)(25,41,28,42)(29,39,35,40)(36,43,38,44);(6,9)(7,11)(13,15,19,26,20,27,18,34)(14,21,23,24,17,31,33,37)(16,28,29,43,30,25,35,44)(22,38,40,42,32,36,39,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,30)(14,22,17,32)(15,25,27,28)(18,35,19,29)(21,36,31,38)(23,40,33,39)(24,41,37,42)(26,44,34,43);(6,9)(10,12)(13,17,20,14)(15,24,27,37)(16,22,30,32)(18,33,19,23)(21,34,31,26)(25,42,28,41)(29,39,35,40)(36,44,38,43);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,17,33)(15,26,27,34)(16,29,30,35)(21,37,31,24)(22,39,32,40)(25,43,28,44)(36,41,38,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=h,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, 7050985);