#7608375

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1699 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,31,24)(20,25,34,40)(21,26,37,45)(23,28,36,44)(30,47,38,39)(32,46,33,41)(35,42,48,43);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,24,27,29)(20,30,34,38)(21,35,37,48)(23,33,36,32)(25,39,40,47)(26,43,45,42)(28,46,44,41);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,22,34)(18,25,27,40)(19,30,31,38)(21,33,37,32)(23,35,36,48)(24,39,29,47)(26,41,45,46)(28,42,44,43);(10,14)(13,16)(17,21,36,30,22,37,23,38)(18,26,44,47,27,45,28,39)(19,20,33,48,31,34,32,35)(24,40,46,42,29,25,41,43);(2,5)(6,8)(18,27)(20,34)(21,37)(24,29)(28,44)(30,38)(35,48)(41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,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, 7515571);