#9851263

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	#49688	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3966 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9758459);