#6463679

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1774 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,26,22,28,21,27)(19,29,32,24,31,44,33,40)(20,34,36,42,38,25,37,43)(30,46,48,41,35,45,47,39);(2,5)(6,7)(10,14)(11,12)(17,19)(18,24)(20,35)(21,32)(22,31)(23,33)(25,41)(26,29)(27,44)(28,40)(30,38)(34,39)(36,48)(37,47)(42,45)(43,46);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20,22,38)(18,25,28,34)(19,30,31,35)(21,37,23,36)(24,39,40,41)(26,43,27,42)(29,45,44,46)(32,48,33,47);(9,12,13,11)(10,15,14,16)(19,31)(24,40)(29,44)(30,35)(32,33)(39,41)(45,46)(47,48);(17,21,22,23)(18,26,28,27)(19,32,31,33)(20,36,38,37)(24,29,40,44)(25,42,34,43)(30,47,35,48)(39,46,41,45)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370875);