#9007189

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	#54777	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,18,23,26)(19,24,21,22)(20,25,28,32)(27,31,30,29)(33,34,39,40)(35,43,44,41)(36,46,45,42)(37,38,47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,19,23,21)(18,24,26,22)(20,27,28,30)(25,31,32,29)(34,40)(35,44)(37,47)(42,46);(2,6)(3,5)(4,12)(15,16)(19,21)(20,28)(22,24)(25,32)(33,35,39,44)(34,41,40,43)(36,37,45,47)(38,46,48,42);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32)(33,36,39,45)(34,42,40,46)(35,37,44,47)(38,43,48,41);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,21,23,19)(18,22,26,24)(20,27,28,30)(25,31,32,29)(33,37)(34,38)(35,45)(36,44)(39,47)(40,48)(41,46)(42,43);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,22)(18,19)(20,29)(21,26)(23,24)(25,27)(28,31)(30,32)(33,38)(34,37)(35,46)(36,43)(39,48)(40,47)(41,45)(42,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,d^2=h,e^2=gi,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, 8914385);