#8696668

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

4197 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 8603864);