#7312672

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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	#42806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,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, 7219868);