#9017189

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	#48199	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=i,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, 8924385);