#8621084

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4298 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)(3,7)(4,5)(6,8)(9,10,15,18)(11,13,16,21)(12,14,17,23)(19,20,22,24)(25,26,32,35)(27,31,34,40)(28,30,29,33)(36,38,37,39);(2,5)(6,7)(9,11,15,16)(10,13,18,21)(12,19,17,22)(14,20,23,24)(25,27,32,34)(26,31,35,40)(28,37,29,36)(30,39,33,38);(1,2)(3,6)(4,5)(7,8)(11,16)(13,21)(19,22)(20,24)(25,28,32,29)(26,33,35,30)(27,36,34,37)(31,39,40,38);(1,3,4,8)(2,6,5,7)(9,12)(10,14)(11,19)(13,20)(15,17)(16,22)(18,23)(21,24)(25,29,32,28)(26,33,35,30)(27,37,34,36)(31,39,40,38);(9,13,15,21)(10,16,18,11)(12,20,17,24)(14,22,23,19)(25,30,32,33)(26,28,35,29)(27,38,34,39)(31,36,40,37);(9,14)(10,17)(11,20)(12,18)(13,22)(15,23)(16,24)(19,21)(25,31)(26,34)(27,35)(28,39)(29,38)(30,36)(32,40)(33,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8528280);