#8994269

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^2=gi,e^2=h,f^2=g,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, 8901465);