#6496552

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470415
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 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:

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

Polycyclic presentation:

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

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, 6403748);