#8463526

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=g,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8370722);