#8720816

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

4186 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8628012);