#9007253

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

4053 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,e^2=gi,f^2=gh,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, 8914449);