#8649949

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	#51519	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 8557145);