#7489845

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 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)(17,18,24,30,23,29,22,28)(19,31,36,42,35,25,34,43)(20,27,37,44,21,38,39,26)(32,46,47,40,33,41,48,45);(2,5)(3,7)(10,14)(11,12)(17,19,23,35)(18,25,29,31)(20,32,21,33)(22,34,24,36)(26,40,44,45)(27,41,38,46)(28,42,30,43)(37,47,39,48);(9,10)(11,16)(12,15)(13,14)(17,20,23,21)(18,26,29,44)(19,32,35,33)(22,37,24,39)(25,40,31,45)(27,30,38,28)(34,47,36,48)(41,43,46,42);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,27,29,38)(19,33,35,32)(22,37,24,39)(25,41,31,46)(26,30,44,28)(34,47,36,48)(40,43,45,42);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,22,23,24)(18,28,29,30)(19,34,35,36)(20,37,21,39)(25,42,31,43)(26,27,44,38)(32,47,33,48)(40,41,45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfhi,e^b=eh,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, 7397041);