#8934840

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	#51817	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842036);