#8755368

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	#53532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3981 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)(4,7)(5,8)(6,12)(10,14)(11,15)(13,16)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(34,38)(35,43)(36,46)(42,48);(1,3,7,5)(2,8,4,9)(6,13,14,11)(10,16,12,15)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,31,32,30)(33,34,39,38)(35,44,43,40)(36,41,46,47)(37,48,45,42);(2,4)(8,9)(10,12)(15,16)(17,18,21,24)(19,22,27,25)(20,23,29,32)(26,30,28,31)(33,35,39,43)(34,40,38,44)(36,45,46,37)(41,48,47,42);(1,4,7,2)(3,9,5,8)(6,12,14,10)(11,16,13,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,48)(42,44);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,37,39,45)(34,42,38,48)(35,36,43,46)(40,41,44,47);(1,6,7,14)(2,10,4,12)(3,11,5,13)(8,15,9,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)(33,38,39,34)(35,44,43,40)(36,47,46,41)(37,48,45,42)

Polycyclic presentation:

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

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, 8662564);