#8458676

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

4542 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)(3,8,4,7)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26)(27,31)(28,38)(29,33)(30,34)(32,35)(36,40)(37,39);(2,6)(7,8)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,27,32,30)(26,31,35,34)(28,37,29,36)(33,40,38,39);(1,3,5,4)(2,7,6,8)(25,28,32,29)(26,33,35,38)(27,36,30,37)(31,39,34,40);(1,2)(3,7)(4,8)(5,6)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,29,32,28)(26,33,35,38)(27,37,30,36)(31,39,34,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,30,32,27)(26,34,35,31)(28,37,29,36)(33,40,38,39);(1,4,5,3)(2,7,6,8)(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,31,32,34)(26,27,35,30)(28,39,29,40)(33,36,38,37)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=i,f^2=gi,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, 8365872);