#7609624

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1803 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,28,34)(20,24,32,38)(21,25,31,37)(27,33,41,45)(29,35,40,44)(30,36,42,46)(39,43,47,48);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(20,30)(21,31)(24,36)(25,37)(27,39)(29,40)(32,42)(33,43)(35,44)(38,46)(41,47)(45,48);(10,14)(11,15)(17,19,22,28)(18,23,26,34)(20,27,32,41)(21,29,31,40)(24,33,38,45)(25,35,37,44)(30,39,42,47)(36,43,46,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,31,42,22,32,21,30)(18,24,37,46,26,38,25,36)(19,27,40,47,28,41,29,39)(23,33,44,48,34,45,35,43);(2,7)(6,8)(19,28)(20,32)(23,34)(24,38)(29,40)(30,42)(35,44)(36,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fhi,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, 7516820);