#9017471

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	#53704	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=ghi,e^2=hi,f^2=h,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, 8924667);