#9545757

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	#54510	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3840 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=i,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=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9452953);