#8455920

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4554 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=ghi,e^2=i,f^2=g,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, 8363116);