#6848608

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1669 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fh,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6755804);