#8922955

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

4183 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=ghi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8830151);