#7011670

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1385 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=i,d^2=i,e^2=fi,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, 6918866);