#8944420

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 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)(3,4)(5,13)(6,7)(8,14)(9,10)(11,16)(12,15)(17,18,23,27)(19,20,28,24)(21,25,29,32)(22,26,31,30)(33,34)(35,41)(36,40)(37,45)(38,39)(42,46)(43,47)(44,48);(1,3)(2,4)(5,11)(6,9)(7,10)(8,12)(13,16)(14,15)(17,19)(18,24)(20,27)(21,22)(23,28)(25,30)(26,32)(29,31)(34,39)(40,41)(42,45)(47,48);(2,7)(3,9)(5,14)(12,16)(21,29)(22,31)(25,32)(26,30)(33,35,38,36)(34,40,39,41)(37,44,46,43)(42,48,45,47);(1,4)(2,3)(5,12)(6,10)(7,9)(8,11)(13,15)(14,16)(17,20,23,24)(18,19,27,28)(21,30,29,26)(22,25,31,32)(33,36,38,35)(34,41,39,40)(37,44,46,43)(42,48,45,47);(1,2)(3,10)(4,9)(5,8)(6,7)(11,16)(12,15)(13,14)(17,21)(18,25)(19,22)(20,26)(23,29)(24,30)(27,32)(28,31)(33,34)(35,41)(36,40)(37,42)(38,39)(43,48)(44,47)(45,46);(1,5,6,14)(2,8,7,13)(3,11,9,15)(4,12,10,16)(17,22)(18,26)(19,21)(20,25)(23,31)(24,32)(27,30)(28,29)(33,37,38,46)(34,42,39,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851616);