#8772955

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

4226 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8680151);