#9058575

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	#46868	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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)(3,10)(9,14)(11,16)(33,34,39,37)(35,38,44,40)(36,45,47,41)(42,46,43,48);(1,2,6,7)(3,4,10,8)(5,9,13,14)(11,12,16,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,35,39,44)(34,40,37,38)(36,42,47,43)(41,46,45,48);(3,10)(4,8)(5,13)(9,14)(18,22)(19,26)(20,29)(31,32)(33,36,39,47)(34,41,37,45)(35,42,44,43)(38,48,40,46);(1,3,6,10)(2,8,7,4)(5,11,13,16)(9,15,14,12)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,36)(34,41)(35,43)(37,45)(38,46)(39,47)(40,48)(42,44);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,37,39,34)(35,38,44,40)(36,41,47,45)(42,46,43,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38)(34,35)(36,46)(37,44)(39,40)(41,43)(42,45)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965771);