#7312370

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	#53053	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 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)(17,18,23,37,22,27,19,24)(20,30,28,25,33,38,31,41)(21,26,34,42,36,44,29,39)(32,47,45,40,35,48,46,43);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,37)(20,31,33,28)(21,34,36,29)(25,30,41,38)(26,42,44,39)(32,45,35,46)(40,48,43,47);(9,11)(10,13)(12,16)(14,15)(17,20,22,33)(18,25,27,41)(19,28,23,31)(21,35,36,32)(24,38,37,30)(26,43,44,40)(29,46,34,45)(39,47,42,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21,22,36)(18,26,27,44)(19,29,23,34)(20,32,33,35)(24,39,37,42)(25,40,41,43)(28,45,31,46)(30,47,38,48);(2,5)(6,8)(10,14)(13,15)(18,27)(21,36)(24,37)(25,41)(29,34)(30,38)(32,35)(45,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7219566);