#9003920

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	#54264	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3910 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,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, 8911116);