#8621913

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	#46559	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(33,34,38,39)(35,40,46,43)(36,37,42,41)(44,47,48,45);(2,4)(3,10)(8,12)(11,13)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,34,38,39)(35,43,46,40)(36,37,42,41)(44,45,48,47);(2,4)(7,9)(8,12)(15,16)(21,30)(24,28)(26,29)(31,32)(34,39)(36,42)(40,43)(44,48);(1,3,6,10)(2,7,4,9)(5,13,14,11)(8,16,12,15)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,35)(34,40)(36,44)(37,47)(38,46)(39,43)(41,45)(42,48);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,36,38,42)(34,41,39,37)(35,44,46,48)(40,45,43,47);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,19)(18,25)(20,23)(21,26)(22,27)(24,32)(28,31)(29,30)(33,37,38,41)(34,42,39,36)(35,45,46,47)(40,44,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8529109);