#8193717

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8100913);