#8605698

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

4241 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=g,e^2=g,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, 8512894);