#8755086

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	#56189	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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)(3,9,10,7)(4,12,13,8)(11,14,16,15)(17,18)(19,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(2,5)(4,13)(7,9)(11,16)(18,23)(19,26)(24,29)(27,32)(36,44)(39,46)(42,48)(45,47);(3,10)(7,9)(11,16)(14,15)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,31)(30,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3,6,10)(2,7,5,9)(4,11,13,16)(8,14,12,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(1,4,6,13)(2,8,5,12)(3,11,10,16)(7,14,9,15)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,5,6,2)(3,9,10,7)(4,12,13,8)(11,15,16,14)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662282);