#6808780

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
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

1615 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=fi,e^2=hi,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

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, 6715976);