#8618116

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=g,e^2=gi,f^2=g,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, 8525312);