#6629964

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^b=ei,h^d=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537160);