#6828128

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1301 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)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,24,28,35)(20,25,33,36)(22,26,32,37)(27,40,43,38)(29,39,31,42)(30,45,34,41)(44,48,46,47);(1,3,6,2)(4,8,5,7)(18,23)(19,27)(20,30)(22,32)(24,38)(25,41)(28,43)(33,34)(35,40)(36,45)(39,42)(44,46);(10,13)(14,16)(17,19,22,27,21,28,32,43)(18,24,26,40,23,35,37,38)(20,31,34,46,33,29,30,44)(25,42,41,47,36,39,45,48);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(18,23)(19,28)(25,36)(26,37)(27,43)(29,31)(41,45)(44,46);(1,5,6,4)(2,7,3,8)(17,20,32,30,21,33,22,34)(18,25,37,45,23,36,26,41)(19,29,43,46,28,31,27,44)(24,39,38,47,35,42,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6735324);