#9717762

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

4053 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)(3,4,8,11)(5,13,15,9)(10,14,16,12)(17,18)(19,24)(20,29)(21,26)(22,30)(23,25)(27,32)(28,31)(33,34,39,41)(35,45,46,40)(36,37,43,42)(38,48,47,44);(2,7)(4,11)(9,13)(12,14)(17,19,23,20)(18,24,25,29)(21,30,31,27)(22,28,32,26)(33,35)(34,40)(36,47)(37,48)(38,43)(39,46)(41,45)(42,44);(2,7)(3,8)(9,13)(10,16)(18,25)(19,20)(26,28)(27,30)(34,41)(35,46)(36,43)(44,48);(1,3,6,8)(2,4,7,11)(5,10,15,16)(9,12,13,14)(17,20,23,19)(18,24,25,29)(21,30,31,27)(22,26,32,28)(33,36,39,43)(34,42,41,37)(35,47,46,38)(40,44,45,48);(1,4)(2,8)(3,7)(5,14)(6,11)(9,10)(12,15)(13,16)(17,21,23,31)(18,26,25,28)(19,27,20,30)(22,24,32,29)(33,37,39,42)(34,43,41,36)(35,44,46,48)(38,40,47,45);(1,5,6,15)(2,9,7,13)(3,10,8,16)(4,12,11,14)(17,22,23,32)(18,27,25,30)(19,28,20,26)(21,29,31,24)(33,38,39,47)(34,44,41,48)(35,43,46,36)(37,40,42,45)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9624958);