#8993102

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
prime radical	#42073	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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)(3,11,12,7)(4,13,14,9)(5,10,15,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,31)(29,30)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(1,3,6,12)(2,7,8,11)(4,5,14,15)(9,10,13,16)(19,26)(23,25)(27,28)(31,32)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(2,8)(4,14)(5,15)(7,11)(17,19,22,26)(18,23,21,25)(20,28,30,27)(24,32,29,31)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,4,6,14)(2,9,8,13)(3,5,12,15)(7,10,11,16)(17,20,22,30)(18,24,21,29)(19,27,26,28)(23,31,25,32)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,5)(2,10)(3,4)(6,15)(7,9)(8,16)(11,13)(12,14)(17,19)(18,25)(20,28)(21,23)(22,26)(24,31)(27,30)(29,32)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,12)(13,14)(15,16)(17,21,22,18)(19,23,26,25)(20,29,30,24)(27,31,28,32)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,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, 8900298);