#8733958

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

4261 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8641154);