#8994263

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4046 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=i,e^2=h,f^2=g,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, 8901459);