#6529472

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

2729 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,g^c=gi,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, 6436668);