#6833148

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1217 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)(9,10,12,13)(11,14,15,16)(17,18,22,27)(19,24,30,39)(20,32,35,25)(21,26,31,40)(23,28,34,42)(29,45,47,38)(33,44,37,41)(36,48,46,43);(1,2)(3,6)(4,5)(7,8)(17,19,22,30)(18,24,27,39)(20,33,35,37)(21,34,31,23)(25,41,32,44)(26,42,40,28)(29,46,47,36)(38,48,45,43);(10,13)(14,16)(17,20,34,46,22,35,23,36)(18,25,42,48,27,32,28,43)(19,29,21,37,30,47,31,33)(24,38,26,44,39,45,40,41);(1,3,4,7)(2,6,5,8)(20,35)(25,32)(29,47)(33,37)(36,46)(38,45)(41,44)(43,48);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,23,19,22,31,34,30)(18,26,28,24,27,40,42,39)(20,33,36,47,35,37,46,29)(25,41,43,45,32,44,48,38)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=f,e^2=ghi,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, 6740344);