#6466689

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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)(11,14)(12,16)(13,15)(17,18)(19,24)(20,25)(21,23)(22,26)(27,38)(28,37)(29,33)(30,40)(31,35)(32,34)(36,42)(39,44)(41,47)(43,45)(46,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(18,23)(19,27)(20,30)(22,31)(24,36)(25,39)(28,43)(29,42)(32,44)(33,38)(34,40)(46,47);(9,12,13,11)(10,14,15,16)(17,19,22,27,21,29,31,42)(18,24,26,38,23,33,35,36)(20,28,30,41,32,43,44,48)(25,37,40,47,34,45,39,46);(17,20,31,44,21,32,22,30)(18,25,35,39,23,34,26,40)(19,28,42,48,29,43,27,41)(24,37,36,46,33,45,38,47);(1,3,4,7)(2,6,5,8)(19,29)(24,33)(27,42)(28,43)(36,38)(37,45)(41,48)(46,47)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,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, 6373885);