#6800822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 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)(17,18)(19,25)(20,35)(21,27)(22,29)(23,28)(24,30)(26,36)(31,45)(32,40)(33,46)(34,42)(37,44)(38,41)(39,48)(43,47);(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(17,19,24,34)(18,25,30,42)(20,32,21,31)(22,38,23,33)(26,43,44,39)(27,45,35,40)(28,46,29,41)(36,47,37,48);(2,5)(6,8)(10,13)(14,15)(17,20,24,21)(18,26,30,44)(19,31,34,32)(22,36,23,37)(25,39,42,43)(27,28,35,29)(33,47,38,48)(40,46,45,41);(1,3,4,7)(2,6,5,8)(17,21,24,20)(18,27,30,35)(19,32,34,31)(22,36,23,37)(25,40,42,45)(26,28,44,29)(33,47,38,48)(39,46,43,41);(9,11,12,16)(10,14,13,15)(17,22,24,23)(18,28,30,29)(19,33,34,38)(20,36,21,37)(25,41,42,46)(26,35,44,27)(31,47,32,48)(39,40,43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6708018);