#8621922

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

4174 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8529118);