#8718067

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8625263);