#8924995

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

4161 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(19,25)(23,30)(26,28)(31,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,47)(44,48);(2,7)(4,8)(10,14)(12,15)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(34,39)(36,40)(41,45)(44,47);(5,13)(10,14)(11,16)(12,15)(18,22)(23,30)(24,27)(31,32)(33,34,38,39)(35,40,43,36)(37,41,46,45)(42,47,48,44);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)

Polycyclic presentation:

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

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, 8832191);