#8902506

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

4124 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8809702);