#6617572

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(13,14,19,31,18,22,17,23)(15,24,26,20,28,35,27,34)(16,21,32,42,29,37,30,36)(25,38,40,33,39,43,41,44);(6,10)(8,11)(13,15)(14,20)(16,25)(17,26)(18,28)(19,27)(21,33)(22,34)(23,35)(24,31)(29,39)(30,40)(32,41)(36,43)(37,44)(38,42);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,21)(15,25)(17,30)(18,29)(19,32)(20,33)(22,37)(23,36)(24,38)(26,40)(27,41)(28,39)(31,42)(34,44)(35,43);(5,7,9,12)(6,11,10,8)(14,22)(16,29)(20,34)(23,31)(24,35)(25,39)(30,32)(40,41);(5,8,9,11)(6,7,10,12)(13,17,18,19)(14,23,22,31)(15,26,28,27)(16,30,29,32)(20,35,34,24)(21,36,37,42)(25,40,39,41)(33,43,44,38)

Polycyclic presentation:

2^9|a^2=fi,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524768);