#8622297

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

4111 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)(3,10,5,7)(4,8,13,16)(11,15,12,14)(17,18)(19,27)(20,29)(21,28)(22,26)(23,24)(25,31)(30,32)(33,34)(35,40)(36,41)(37,39)(38,42)(43,46)(44,48)(45,47);(1,3,6,5)(2,7,9,10)(4,12,13,11)(8,15,16,14)(17,19,23,28)(18,21,24,27)(20,30,31,22)(25,32,29,26)(33,35,39,42)(34,40,37,38)(36,44,47,43)(41,48,45,46);(4,13)(8,16)(11,12)(14,15)(18,24)(21,27)(25,29)(26,32)(34,37)(35,42)(41,45)(43,44);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,20,23,31)(18,25,24,29)(19,22,28,30)(21,26,27,32)(33,36,39,47)(34,41,37,45)(35,43,42,44)(38,48,40,46);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,16)(10,15)(17,21)(18,19)(20,26)(22,25)(23,27)(24,28)(29,30)(31,32)(33,37,39,34)(35,40,42,38)(36,45,47,41)(43,46,44,48);(1,5,6,3)(2,7,9,10)(4,12,13,11)(8,14,16,15)(17,22)(18,26)(19,20)(21,25)(23,30)(24,32)(27,29)(28,31)(33,38,39,40)(34,42,37,35)(36,46,47,48)(41,43,45,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=h,e^2=i,f^2=gh,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, 8529493);