#8656197

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

4163 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=hi,e^2=hi,f^2=gi,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, 8563393);