#6573421

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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

3074 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)(13,14,18,24,19,23,17,22)(15,25,27,34,28,38,32,20)(16,21,29,35,30,36,31,37)(26,39,41,44,42,43,40,33);(5,6,10,9)(7,12,8,11)(13,15)(14,20)(16,26)(17,27)(18,32)(19,28)(21,33)(22,25)(23,34)(24,38)(29,40)(30,42)(31,41)(35,43)(36,44)(37,39);(2,4)(5,7,10,8)(6,11,9,12)(13,16,19,30)(14,21,23,36)(15,26,28,42)(17,31,18,29)(20,33,34,44)(22,37,24,35)(25,39,38,43)(27,41,32,40);(5,8,10,7)(6,11,9,12)(13,17,19,18)(14,22,23,24)(15,27,28,32)(16,29,30,31)(20,25,34,38)(21,35,36,37)(26,40,42,41)(33,43,44,39);(5,9,10,6)(7,12,8,11)(14,23)(20,34)(21,36)(22,24)(25,38)(33,44)(35,37)(39,43)

Polycyclic presentation:

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

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, 6480617);