#9015530

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	#54923	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4233 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=hi,e^2=i,f^2=h,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, 8922726);