#8761969

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

4254 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,21)(20,25)(22,23)(24,27)(26,29)(28,32)(30,31)(33,34,38,41)(35,39,44,40)(36,37,46,48)(42,43,45,47);(2,6)(4,13)(7,9)(11,16)(18,23)(21,24)(25,31)(29,32)(33,35)(34,39)(36,45)(37,47)(38,44)(40,41)(42,46)(43,48);(2,6)(3,10)(8,12)(11,16)(17,19,22,27)(18,21,23,24)(20,28,30,26)(25,32,31,29)(33,35,38,44)(34,40,41,39)(36,45,46,42)(37,43,48,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19)(18,24)(20,26)(21,23)(22,27)(25,32)(28,30)(29,31)(33,35)(34,39)(36,42)(37,43)(38,44)(40,41)(45,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,25)(19,26)(21,29)(22,30)(23,31)(24,32)(27,28)(33,36)(34,37)(35,42)(38,46)(39,43)(40,47)(41,48)(44,45);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,24)(18,19,23,27)(20,29,30,32)(25,26,31,28)(33,37)(34,36)(35,43)(38,48)(39,42)(40,45)(41,46)(44,47)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8669165);