#7009550

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

1253 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,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, 6916746);