#8227353

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

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

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, 8134549);