#6726229

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

1269 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)(11,14)(12,15)(13,16)(17,18,22,25)(19,29,32,24)(20,34,36,26)(21,27,23,28)(30,38,33,42)(31,39,45,43)(35,41,37,40)(44,48,46,47);(2,5)(3,7)(9,11,13,12)(10,14,16,15)(17,19)(18,24)(20,31)(21,33)(22,32)(23,30)(25,29)(26,39)(27,38)(28,42)(34,43)(35,44)(36,45)(37,46)(40,48)(41,47);(2,5)(6,8)(9,11,13,12)(10,15,16,14)(18,25)(19,30)(21,23)(24,38)(26,34)(29,42)(31,44)(32,33)(35,37)(39,47)(43,48)(45,46);(2,5)(6,8)(17,20)(18,26)(19,31)(21,37)(22,36)(23,35)(24,39)(25,34)(27,41)(28,40)(29,43)(30,44)(32,45)(33,46)(38,47)(42,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,21)(18,27)(19,32)(20,35)(22,23)(24,29)(25,28)(26,40)(34,41)(36,37)(44,46)(47,48)

Polycyclic presentation:

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

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, 6633425);