#8901350

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	#40403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808546);