#8993198

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

4134 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=gi,f^2=h,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, 8900394);