#6675527

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1685 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)(3,10)(4,13)(5,8)(6,9)(7,12)(11,15)(14,16)(17,18,23,28)(19,24,32,40)(20,26,35,44)(21,25,22,27)(29,38,31,39)(30,37,46,47)(33,42,45,48)(34,41,36,43);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,24)(20,33)(21,31)(22,29)(23,32)(25,39)(26,42)(27,38)(28,40)(30,34)(35,45)(36,46)(37,41)(43,47)(44,48);(18,25)(19,29)(21,22)(26,41)(27,28)(30,33)(31,32)(34,36)(38,39)(42,48)(43,44)(45,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,20,23,35)(18,26,28,44)(19,30,32,46)(21,34,22,36)(24,37,40,47)(25,41,27,43)(29,33,31,45)(38,42,39,48)

Polycyclic presentation:

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

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, 6582723);