#7592750

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1291 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,23,27)(19,28,31,24)(20,32,36,48)(21,37,30,40)(22,38,34,26)(25,41,43,35)(29,45,47,42)(33,44,46,39);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,24,27,28)(20,33,36,46)(21,22,30,34)(25,42,43,45)(26,40,38,37)(29,41,47,35)(32,39,48,44);(17,20,34,41,23,36,22,35)(18,25,38,48,27,43,26,32)(19,29,30,46,31,47,21,33)(24,39,40,45,28,44,37,42);(11,15)(13,16)(17,21,22,31,23,30,34,19)(18,24,26,37,27,28,38,40)(20,33,35,47,36,46,41,29)(25,39,32,42,43,44,48,45);(1,3,4,8)(2,6,5,7)(17,22,23,34)(18,26,27,38)(19,30,31,21)(20,35,36,41)(24,40,28,37)(25,32,43,48)(29,46,47,33)(39,45,44,42)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7499946);