#8197263

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	#46731	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104459);