#6485553

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1393 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6392749);