#8739336

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

4016 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18,22,25)(19,24,28,23)(20,29,30,21)(26,27,32,31)(35,37)(36,40)(43,45)(44,46);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,19,22,28)(18,23,25,24)(20,26,30,32)(21,27,29,31)(33,34,39,42)(35,36,37,40)(38,41,47,48)(43,44,45,46);(4,12)(8,15)(11,13)(14,16)(17,19,22,28)(18,23,25,24)(20,26,30,32)(21,27,29,31)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,46,48,44);(1,4,6,12)(2,8,9,15)(3,11,5,13)(7,14,10,16)(17,19)(18,24)(20,26)(21,31)(22,28)(23,25)(27,29)(30,32)(33,36,39,40)(34,35,42,37)(38,46,47,44)(41,45,48,43);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,20,22,30)(18,21,25,29)(19,26,28,32)(23,27,24,31)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,48,46);(1,5,6,3)(2,7,9,10)(4,11,12,13)(8,16,15,14)(17,21)(18,20)(19,27)(22,29)(23,26)(24,32)(25,30)(28,31)(33,38,39,47)(34,41,42,48)(35,43,37,45)(36,44,40,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8646532);