#6496556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
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	#3006	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1632 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)(18,23)(19,27)(21,33)(25,35)(26,36)(28,41)(29,30)(32,44)(34,38)(40,46)(42,43)(45,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,18,21,23,22,26,33,36)(19,24,29,34,30,39,27,38)(20,25,32,35,31,40,44,46)(28,37,43,45,42,47,41,48);(3,9)(6,13)(10,15)(14,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,34)(25,37)(26,39)(27,33)(31,42)(32,43)(35,45)(36,38)(40,47)(41,44)(46,48);(3,9)(6,13)(10,15)(14,16)(17,20)(18,25)(19,28)(21,32)(22,31)(23,35)(24,37)(26,40)(27,41)(29,43)(30,42)(33,44)(34,45)(36,46)(38,48)(39,47);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(20,31)(25,40)(28,42)(32,44)(35,46)(37,47)(41,43)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=f,e^2=h,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,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, 6403752);