#9065947

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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8973143);