#6529912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
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	C₂×#3122	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1662 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=h,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,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, 6437108);