#9066128

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

4170 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)(20,30)(21,26)(22,24)(23,25)(28,31)(29,32)(33,34,39,43)(35,40,44,46)(36,38,42,48)(37,41,47,45);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19,23,22)(18,24,25,27)(20,28,26,32)(21,31,30,29)(33,35)(34,40)(36,45)(37,38)(39,44)(41,42)(43,46)(47,48);(4,11)(7,9)(12,16)(14,15)(18,25)(21,30)(24,27)(29,31)(35,44)(37,47)(40,46)(41,45);(1,3,6,2)(4,7,11,9)(5,8,13,10)(12,15,16,14)(17,20)(18,21)(19,28)(22,32)(23,26)(24,31)(25,30)(27,29)(33,36)(34,38)(35,45)(37,40)(39,42)(41,44)(43,48)(46,47);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21)(18,26)(19,29)(20,25)(22,31)(23,30)(24,28)(27,32)(33,37)(34,41)(35,38)(36,46)(39,47)(40,42)(43,45)(44,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,19)(18,27,25,24)(20,28,26,32)(21,31,30,29)(33,38)(34,42)(35,37)(36,43)(39,48)(40,41)(44,47)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=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, 8973324);