#7008067

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1329 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6915263);