#9017380

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	#52311	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924576);