#6813398

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1305 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)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,26)(21,27)(22,37)(23,29)(24,30)(28,38)(31,39)(32,40)(33,41)(34,42)(35,44)(36,45)(43,47)(46,48);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19)(18,25)(20,33)(21,22)(23,38)(24,32)(26,41)(27,37)(28,29)(30,40)(31,35)(34,46)(36,47)(39,44)(42,48)(43,45);(10,14)(13,16)(17,20,21,34,24,36,23,35)(18,26,27,42,30,45,29,44)(19,31,38,47,32,46,22,33)(25,39,28,43,40,48,37,41);(2,5)(6,8)(17,21,24,23)(18,27,30,29)(19,22,32,38)(20,34,36,35)(25,37,40,28)(26,42,45,44)(31,33,46,47)(39,41,48,43);(1,3,4,7)(2,6,5,8)(17,22,21,19,24,38,23,32)(18,28,27,40,30,37,29,25)(20,33,34,31,36,47,35,46)(26,43,42,48,45,41,44,39)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=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, 6720594);