#8747935

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	#2819	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3873 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18,22,21)(19,24,27,23)(20,28,29,25)(26,30,32,31)(33,34)(35,38)(36,41)(37,42)(39,43)(40,45)(44,46)(47,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,22,27)(18,23,21,24)(20,26,29,32)(25,30,28,31)(34,35)(36,43)(40,42)(44,48);(2,3)(7,9)(8,10)(15,16)(17,19,22,27)(18,23,21,24)(20,26,29,32)(25,30,28,31)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,30)(29,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20,22,29)(18,25,21,28)(19,26,27,32)(23,30,24,31)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,18)(19,24,27,23)(20,25,29,28)(26,30,32,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=g,d^2=gh,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, 8655131);