#6695042

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25)(19,23,29,39)(20,31,33,24)(22,26,35,43)(27,45,46,37)(28,38,30,40)(32,48,47,41)(34,44,36,42);(1,3)(2,5)(4,8)(6,7)(17,19,21,29)(18,23,25,39)(20,32,33,47)(22,28,35,30)(24,41,31,48)(26,38,43,40)(27,36,46,34)(37,44,45,42);(10,13)(14,16)(17,20,21,33)(18,24,25,31)(19,27,29,46)(22,36,35,34)(23,37,39,45)(26,44,43,42)(28,32,30,47)(38,41,40,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(20,33)(24,31)(27,46)(32,47)(34,36)(37,45)(41,48)(42,44);(3,8)(5,7)(19,28)(22,35)(23,38)(26,43)(27,32)(29,30)(34,36)(37,41)(39,40)(42,44)(45,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6602238);