#6529915

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×#3119	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1627 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=h,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,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, 6437111);