#8534433

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

4264 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)(3,9)(4,6)(5,8)(7,11)(10,15)(12,13)(14,16)(17,18)(19,24)(20,26)(21,31)(22,27)(23,25)(28,30)(29,32)(33,34)(35,42)(36,46)(37,41)(38,39)(40,45)(43,44)(47,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,19)(18,24)(20,25)(21,28)(22,29)(23,26)(27,32)(30,31)(34,39)(36,41)(40,42)(43,47);(5,13)(8,12)(10,16)(14,15)(18,25)(19,26)(21,27)(29,30)(33,35,38,45)(34,40,39,42)(36,43,41,47)(37,44,46,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20,23,24)(18,26,25,19)(21,30,27,29)(22,32,31,28)(33,36)(34,37)(35,43)(38,41)(39,46)(40,44)(42,48)(45,47);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,23,27)(18,22,25,31)(19,28,26,32)(20,30,24,29)(33,35)(34,40)(36,43)(37,44)(38,45)(39,42)(41,47)(46,48);(1,3)(2,9)(4,7)(5,10)(6,11)(8,15)(12,14)(13,16)(17,22)(18,27)(19,29)(20,28)(21,25)(23,31)(24,32)(26,30)(33,37)(34,41)(35,44)(36,39)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

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

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, 8441629);