#8756278

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

4162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8663474);