#9712418

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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)(3,5,9,15)(4,8,13,16)(10,14,11,12)(17,18,22,25)(19,26,23,20)(21,24,30,32)(27,31,29,28)(33,34)(35,41)(36,40)(37,42)(38,43)(39,46)(44,45)(47,48);(1,3)(2,5)(4,11)(6,9)(7,15)(8,12)(10,13)(14,16)(17,19)(18,20)(21,29)(22,23)(24,31)(25,26)(27,30)(28,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,43)(42,48);(2,7)(3,9)(8,16)(10,11)(17,20,22,26)(18,23,25,19)(21,28,30,31)(24,29,32,27)(33,36,38,42)(34,40,43,37)(35,44,46,48)(39,47,41,45);(1,2)(3,5)(4,8)(6,7)(9,15)(10,14)(11,12)(13,16)(17,20)(18,19)(21,28)(22,26)(23,25)(24,27)(29,32)(30,31);(1,4,6,13)(2,8,7,16)(3,10,9,11)(5,14,15,12)(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,32)(26,31)(33,35)(34,41)(36,44)(37,47)(38,46)(39,43)(40,45)(42,48);(1,5)(2,3)(4,12)(6,15)(7,9)(8,11)(10,16)(13,14)(17,19)(18,20)(21,29)(22,23)(24,31)(25,26)(27,30)(28,32)(33,37,38,40)(34,42,43,36)(35,45,46,47)(39,48,41,44)

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=i,f^2=ghi,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, 9619614);