#8824404

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731600);