#8650974

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

4113 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=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, 8558170);