#8696731

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

4239 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603927);