#8992922

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
prime radical	#51241	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8900118);