#8661367

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

4145 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8568563);