#8445784

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	#52375	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4425 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8352980);