#8724806

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=hi,e^2=gh,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, 8632002);