#8270587

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	#44909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,2,7,6)(3,10,4,8)(5,13,14,9)(11,15,12,16)(17,19,23,22)(18,24,27,28)(20,29,21,30)(25,31,26,32)(34,38)(39,41)(40,45)(46,47);(1,3,7,4)(2,8,6,10)(5,11,14,12)(9,15,13,16)(17,20,23,21)(18,25,27,26)(19,29,22,30)(24,31,28,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4,7,3)(2,8,6,10)(5,12,14,11)(9,15,13,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5,7,14)(2,9,6,13)(3,11,4,12)(8,15,10,16)(17,21,23,20)(18,26,27,25)(19,29,22,30)(24,31,28,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47);(1,6,7,2)(3,8,4,10)(5,13,14,9)(11,15,12,16)(17,22,23,19)(18,24,27,28)(20,29,21,30)(25,32,26,31)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=i,e^2=hi,f^2=gi,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, 8177783);