#8696669

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

4193 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:

(2,4)(3,9)(8,12)(11,13)(17,18,23,29)(19,24,31,32)(20,26,30,25)(21,22,28,27)(34,39)(40,41)(42,45)(47,48);(1,2,6,4)(3,7,9,10)(5,8,14,12)(11,15,13,16)(17,19)(18,24)(20,27)(21,26)(22,30)(23,31)(25,28)(29,32)(33,34,38,39)(35,40,36,41)(37,42,46,45)(43,47,44,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,20)(18,25)(19,22)(21,24)(23,30)(26,29)(27,31)(28,32)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,3)(2,7)(4,10)(5,13)(6,9)(8,16)(11,14)(12,15)(17,20)(18,26)(19,27)(21,24)(22,31)(23,30)(25,29)(28,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,48)(44,47);(1,4,6,2)(3,10,9,7)(5,8,14,12)(11,15,13,16)(17,21)(18,27)(19,26)(20,24)(22,29)(23,28)(25,31)(30,32)(33,36,38,35)(34,41,39,40)(37,43,46,44)(42,47,45,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22,23,27)(18,28,29,21)(19,30,31,20)(24,25,32,26)(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=g,c^2=ghi,e^2=g,f^2=gh,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, 8603865);