#7003309

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1910 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,28,44)(16,21,27,40)(17,22,26,42)(18,33,35,23)(20,36,39,29)(30,38,31,41)(32,43,34,37);(5,6)(7,11)(8,10)(9,12)(13,15,18,29,19,28,35,36)(14,20,23,25,24,39,33,44)(16,26,30,34,27,17,31,32)(21,37,41,42,40,43,38,22);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,27)(14,21,24,40)(15,26,28,17)(18,30,35,31)(20,37,39,43)(22,25,42,44)(23,41,33,38)(29,34,36,32);(7,12)(9,11)(13,17,19,26)(14,22,24,42)(15,27,28,16)(18,34,35,32)(20,38,39,41)(21,25,40,44)(23,43,33,37)(29,30,36,31);(6,10)(9,11)(15,28)(16,27)(20,39)(21,40)(25,44)(29,36)(30,31)(38,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,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, 6910505);