#9164010

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

4193 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9071206);