#6529585

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500226
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(17,18,23,35,22,25,21,27)(19,28,34,24,31,41,30,37)(20,26,36,46,32,39,33,40)(29,42,45,38,43,48,44,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,34)(22,31)(23,30)(25,37)(26,38)(27,41)(28,35)(32,43)(33,45)(36,44)(39,47)(40,48)(42,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(18,25)(21,23)(24,37)(26,39)(30,34)(33,36)(38,47)(44,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,26)(19,29)(21,33)(22,32)(23,36)(24,38)(25,39)(27,40)(28,42)(30,44)(31,43)(34,45)(35,46)(37,47)(41,48);(2,8)(6,9)(7,16)(14,15)(20,32)(26,39)(29,43)(33,36)(38,47)(40,46)(42,48)(44,45)

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=h,f^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=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, 6436781);