#8649933

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4038 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)(19,25)(20,30)(21,22)(23,26)(24,27)(28,32)(29,31)(33,34)(35,42)(36,46)(37,45)(38,40)(39,41)(43,47)(44,48);(1,2,6,9)(3,10,4,7)(5,8,13,16)(11,15,12,14)(17,19,24,29)(18,25,27,31)(20,26,21,28)(22,32,30,23)(33,35,40,45)(34,37,38,42)(36,43,39,44)(41,47,46,48);(1,3,6,4)(2,7,9,10)(5,11,13,12)(8,14,16,15)(17,20,24,21)(18,22,27,30)(19,28,29,26)(23,31,32,25)(33,36,40,39)(34,41,38,46)(35,43,45,44)(37,47,42,48);(1,4,6,3)(2,7,9,10)(5,11,13,12)(8,15,16,14)(17,21,24,20)(18,22,27,30)(19,26,29,28)(23,31,32,25)(33,37)(34,35)(36,47)(38,45)(39,48)(40,42)(41,43)(44,46);(1,2)(3,10)(4,7)(5,8)(6,9)(11,15)(12,14)(13,16)(17,22)(18,20)(19,23)(21,27)(24,30)(25,28)(26,31)(29,32)(33,38,40,34)(35,37,45,42)(36,41,39,46)(43,48,44,47);(1,5,6,13)(2,8,9,16)(3,11,4,12)(7,14,10,15)(17,23)(18,26)(19,22)(20,31)(21,25)(24,32)(27,28)(29,30)(33,39,40,36)(34,41,38,46)(35,44,45,43)(37,47,42,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8557129);