#8691864

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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=g,f^2=h,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, 8599060);