#8696266

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

4184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603462);