#8496895

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

4494 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)(3,7)(4,6)(5,8)(10,11)(12,19)(16,18)(20,24)(25,26,30,29)(27,32,34,31)(28,33,36,37)(35,39,40,38);(1,2,4,6)(3,5,8,7)(9,10,14,11)(12,17,19,15)(13,16,21,18)(20,23,24,22)(27,34)(28,36)(31,32)(33,37);(1,2)(3,5)(4,6)(7,8)(25,27)(26,31)(28,35)(29,32)(30,34)(33,38)(36,40)(37,39);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,40)(37,38);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,36)(26,33,29,37)(27,35,34,40)(31,38,32,39);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,26)(27,32,34,31)(28,37,36,33)(35,39,40,38)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8404091);