#6800804

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=f,e^2=fh,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, 6708000);