#8718344

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	#43360	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625540);