#8873731

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

4133 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)(4,9,12,7)(5,10,13,8)(11,14,16,15)(17,18)(19,25)(20,28)(21,24)(22,26)(23,30)(27,31)(29,32)(33,34)(35,41)(36,43)(37,45)(38,40)(39,47)(42,44)(46,48);(1,2,6,3)(4,9,12,7)(5,10,13,8)(11,14,16,15)(17,18,21,24)(19,22,26,25)(20,23,30,28)(27,31,29,32)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(2,3)(7,9)(8,10)(14,15)(19,26)(22,25)(27,29)(31,32)(33,34,38,40)(35,36,43,41)(37,39,47,45)(42,44,46,48);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,19,21,26)(18,22,24,25)(20,29,30,27)(23,32,28,31)(33,36,38,41)(34,35,40,43)(37,44,47,48)(39,42,45,46);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20)(18,23)(19,27)(21,30)(22,31)(24,28)(25,32)(26,29)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,48)(46,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,31)(28,30)(29,32)(33,37)(34,39)(35,42)(36,44)(38,47)(40,45)(41,48)(43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780927);