#8393659

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

4118 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300855);