#8429376

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	#42629	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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)(3,10)(9,15)(11,13)(17,18,22,21)(19,24,26,23)(20,28,29,25)(27,30,32,31)(34,39)(35,36)(42,45)(43,44);(1,2,6,7)(3,8,10,4)(5,9,14,15)(11,16,13,12)(19,26)(23,24)(27,32)(30,31)(33,34,38,39)(35,40,36,41)(37,42,46,45)(43,47,44,48);(17,19)(18,23)(20,27)(21,24)(22,26)(25,30)(28,31)(29,32)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,3)(2,8)(4,7)(5,13)(6,10)(9,12)(11,14)(15,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,48)(44,47);(1,4,6,8)(2,3,7,10)(5,12,14,16)(9,11,15,13)(17,20)(18,25)(19,27)(21,28)(22,29)(23,30)(24,31)(26,32)(33,36,38,35)(34,41,39,40)(37,44,46,43)(42,48,45,47);(1,5,6,14)(2,9,7,15)(3,11,10,13)(4,12,8,16)(17,21,22,18)(19,24,26,23)(20,28,29,25)(27,31,32,30)(33,37,38,46)(34,42,39,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8336572);