#9009236

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3922 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:

(2,7)(3,9)(10,14)(11,16)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,45,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)(33,35,39,37)(34,40,44,42)(36,38,47,45)(41,43,46,48);(3,9)(4,8)(5,13)(10,14)(18,22)(19,25)(20,27)(31,32)(33,36,39,47)(34,41,44,46)(35,45,37,38)(40,48,42,43);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,19,21,25)(18,23,22,29)(20,28,27,26)(24,32,30,31)(33,37,39,35)(34,42,44,40)(36,38,47,45)(41,43,46,48);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,20,21,27)(18,24,22,30)(19,26,25,28)(23,31,29,32)(33,38)(34,43)(35,36)(37,47)(39,45)(40,41)(42,46)(44,48);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)(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=h,c^2=h,d^2=i,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, 8916432);