#6675547

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
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

1663 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)(19,30)(20,34)(21,39)(22,27)(23,41)(24,29)(25,32)(26,37)(28,31)(33,44)(35,47)(36,48)(38,46)(40,43)(42,45);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,25)(20,35)(21,37)(22,32)(23,33)(24,42)(26,28)(27,30)(29,44)(31,38)(34,43)(36,40)(39,46)(41,45)(47,48);(17,20,22,36)(18,26,27,46)(19,31,32,21)(23,38,24,37)(25,43,30,47)(28,45,39,44)(29,48,41,34)(33,40,42,35);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(18,27)(25,30)(26,46)(28,39)(29,41)(34,48)(43,47)(44,45);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,21)(18,28)(19,20)(22,31)(23,40)(24,35)(25,34)(26,45)(27,39)(29,43)(30,48)(32,36)(33,37)(38,42)(41,47)(44,46)

Polycyclic presentation:

2^9|c^2=i,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582743);