#6513102

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500120
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
prime radical	C₂×#3547	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3028 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)(13,14,18,24,19,22,17,23)(15,25,27,34,28,38,32,20)(16,21,29,36,30,35,31,37)(26,39,41,43,42,44,40,33);(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,22,34)(16,26,30,42)(17,27,18,32)(21,33,35,43)(23,25,24,38)(29,40,31,41)(36,44,37,39);(2,4)(6,9)(7,11)(13,16)(14,21)(15,26)(17,31)(18,29)(19,30)(20,33)(22,35)(23,37)(24,36)(25,39)(27,41)(28,42)(32,40)(34,43)(38,44);(5,7,8,11)(6,10,9,12)(14,22)(20,34)(21,35)(23,24)(25,38)(33,43)(36,37)(39,44);(6,9)(10,12)(13,17,19,18)(14,23,22,24)(15,27,28,32)(16,29,30,31)(20,25,34,38)(21,36,35,37)(26,40,42,41)(33,44,43,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6420298);