#9065953

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

4183 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,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, 8973149);