#9062141

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	#47809	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(3,11)(4,15)(5,10)(6,7)(8,14)(9,13)(12,16)(17,18,23,27)(19,28,30,24)(20,29,31,25)(21,22,26,32)(33,34)(35,40)(36,42)(37,43)(38,47)(39,41)(44,46)(45,48);(1,3,7,14)(2,8,6,11)(4,5,13,12)(9,10,15,16)(17,19,23,30)(18,24,27,28)(20,21,31,26)(22,29,32,25)(33,35,39,38)(34,40,41,47)(36,44,45,43)(37,42,46,48);(2,6)(4,13)(5,12)(8,11)(20,31)(21,26)(22,32)(25,29)(34,41)(35,38)(36,45)(37,46);(1,4)(2,9)(3,12)(5,14)(6,15)(7,13)(8,16)(10,11)(17,20,23,31)(18,25,27,29)(19,26,30,21)(22,24,32,28)(33,36,39,45)(34,42,41,48)(35,43,38,44)(37,47,46,40);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,15,11,9)(17,21,23,26)(18,22,27,32)(19,20,30,31)(24,25,28,29)(33,37)(34,43)(35,42)(36,40)(38,48)(39,46)(41,44)(45,47);(1,6,7,2)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,22)(18,26)(19,29)(20,24)(21,27)(23,32)(25,30)(28,31)(33,38,39,35)(34,40,41,47)(36,44,45,43)(37,48,46,42)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969337);