#8726309

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,e^2=ghi,f^2=gi,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, 8633505);