#6700155

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1755 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6607351);