#6529897

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500474
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

1638 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)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,37)(30,42)(31,41)(33,40)(35,39)(36,44)(38,43)(45,47)(46,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,21,32,23,33,22,31)(18,24,27,29,28,41,26,40)(20,35,37,46,36,30,38,47)(25,42,34,48,43,39,44,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20,23,36)(18,25,28,43)(19,30,33,35)(21,37,22,38)(24,39,41,42)(26,44,27,34)(29,45,40,48)(31,46,32,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(20,36)(25,43)(30,35)(34,44)(37,38)(39,42)(45,48)(46,47);(2,8)(6,9)(7,16)(14,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,37,36,38)(24,29,41,40)(25,44,43,34)(30,46,35,47)(39,45,42,48)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=hi,e^2=i,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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6437093);