#9017768

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	#44032	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

(1,2,6,8)(3,7,11,14)(4,9,13,16)(5,12,15,10)(18,23)(19,26)(25,31)(27,29);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,31)(29,30)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(2,8)(3,11)(4,13)(5,15)(19,26)(20,28)(21,24)(25,31)(33,35,39,46)(34,40,43,44)(36,37,42,45)(38,47,48,41);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,13)(15,16)(17,19,22,26)(18,24,23,21)(20,29,28,27)(25,30,31,32)(33,36,39,42)(34,38,43,48)(35,45,46,37)(40,41,44,47);(1,4)(2,9)(3,12)(5,14)(6,13)(7,15)(8,16)(10,11)(17,20,22,28)(18,25,23,31)(19,27,26,29)(21,30,24,32)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,5,6,15)(2,10,8,12)(3,9,11,16)(4,14,13,7)(17,21,22,24)(18,26,23,19)(20,30,28,32)(25,29,31,27)(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=hi,c^2=hi,d^2=ghi,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, 8924964);