#9058446

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	#47129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,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, 8965642);