#9947488

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	#50035	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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)(3,12)(4,10)(5,11)(6,8)(7,9)(13,15)(14,16)(17,18,24,26)(19,20,21,22)(23,31,32,25)(27,30,29,28)(35,41)(38,42)(43,47)(46,48);(1,3,7,6)(2,8,9,12)(4,11,14,13)(5,10,15,16)(17,19)(18,20)(21,24)(22,26)(23,27)(25,28)(29,32)(30,31)(33,34,37,40)(35,42,41,38)(36,39,45,44)(43,48,47,46);(2,9)(4,14)(8,12)(11,13)(17,20,24,22)(18,19,26,21)(23,28,32,30)(25,27,31,29)(33,34)(35,38)(36,39)(37,40)(41,42)(43,46)(44,45)(47,48);(1,4,7,14)(2,10,9,16)(3,13,6,11)(5,12,15,8)(17,21,24,19)(18,20,26,22)(23,29,32,27)(25,28,31,30)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,5)(2,11)(3,10)(4,12)(6,16)(7,15)(8,14)(9,13)(17,22,24,20)(18,19,26,21)(23,28,32,30)(25,29,31,27)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(1,6,7,3)(2,12,9,8)(4,11,14,13)(5,10,15,16)(17,23)(18,25)(19,27)(20,28)(21,29)(22,30)(24,32)(26,31)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gi,e^2=gh,f^2=g,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, 9854684);