#8922606

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	#40111	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3987 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829802);