#8749655

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

3817 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18,23,24)(19,27,21,25)(20,26,30,22)(28,32,31,29)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(18,24)(20,30)(25,27)(28,31)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(4,12)(8,15)(11,13)(14,16)(17,19,23,21)(18,25,24,27)(20,31,30,28)(22,29,26,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,4,6,12)(2,8,9,15)(3,11,5,13)(7,14,10,16)(17,20,23,30)(18,26,24,22)(19,28,21,31)(25,29,27,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,21,23,19)(18,27,24,25)(20,31,30,28)(22,29,26,32)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47);(1,5,6,3)(2,7,9,10)(4,11,12,13)(8,16,15,14)(17,22,23,26)(18,20,24,30)(19,29,21,32)(25,31,27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8656851);