#8227369

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

3846 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=hi,e^2=gi,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, 8134565);