#8692566

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8599762);