#9947373

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

3901 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)(3,8,10,14)(4,5,9,13)(11,12,15,16)(17,18)(19,22)(20,23)(21,26)(24,30)(25,28)(27,32)(29,31)(33,34,39,40)(35,42,37,36)(38,46,47,41)(43,44,45,48);(2,7)(3,10)(4,9)(12,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32)(33,35,39,37)(34,36,40,42)(38,43,47,45)(41,44,46,48);(18,19)(23,25)(24,26)(31,32)(34,40)(36,42)(41,46)(44,48);(1,3)(2,8)(4,11)(5,12)(6,10)(7,14)(9,15)(13,16)(17,19,22,18)(20,25,28,23)(21,26,30,24)(27,32,29,31)(33,36)(34,35)(37,40)(38,44)(39,42)(41,43)(45,46)(47,48);(1,4)(2,5)(3,11)(6,9)(7,13)(8,12)(10,15)(14,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,37,39,35)(34,36,40,42)(38,43,47,45)(41,48,46,44);(1,5)(2,9)(3,12)(4,7)(6,13)(8,15)(10,16)(11,14)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854569);