#8464196

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

4545 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)(9,10)(11,15)(12,22)(13,17)(14,18)(16,19)(20,24)(21,23)(26,29)(28,32)(33,37)(36,38);(1,2,4,6)(3,5,8,7)(9,11,16,14)(10,15,19,18)(12,21,13,20)(17,24,22,23)(25,26,31,29)(27,32,35,28)(30,33,39,37)(34,38,40,36);(1,2)(3,5)(4,6)(7,8)(9,12,16,13)(10,17,19,22)(11,20,14,21)(15,23,18,24)(25,27,31,35)(26,32,29,28)(30,34,39,40)(33,38,37,36);(1,3)(2,5)(4,8)(6,7)(9,13,16,12)(10,17,19,22)(11,21,14,20)(15,23,18,24)(25,28)(26,27)(29,35)(30,38)(31,32)(33,40)(34,37)(36,39);(9,14,16,11)(10,18,19,15)(12,21,13,20)(17,24,22,23)(25,29,31,26)(27,28,35,32)(30,37,39,33)(34,36,40,38);(9,15,16,18)(10,11,19,14)(12,23,13,24)(17,20,22,21)(25,30,31,39)(26,33,29,37)(27,34,35,40)(28,36,32,38)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=h,e^2=gi,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, 8371392);