#6815344

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814827
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

1609 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=f,e^2=h,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,d^c=di,e^c=ei

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, 6722540);