#8993037

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

4068 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^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, 8900233);