#8876373

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	#55109	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8783569);