#7312736

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

1496 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)(20,31,32,25,34,38,29,41)(21,26,30,44,36,43,37,39)(33,47,46,40,35,48,45,42);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,30,36,37)(25,31,41,38)(26,39,43,44)(33,46,35,45)(40,47,42,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,34)(18,25,27,41)(19,29,23,32)(21,35,36,33)(24,38,28,31)(26,42,43,40)(30,46,37,45)(39,47,44,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21,22,36)(18,26,27,43)(19,30,23,37)(20,33,34,35)(24,39,28,44)(25,40,41,42)(29,45,32,46)(31,47,38,48);(2,5)(6,8)(10,14)(13,15)(18,27)(21,36)(24,28)(25,41)(30,37)(31,38)(33,35)(45,46)

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=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, 7219932);