#8437823

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

4412 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,d^2=h,e^2=h,f^2=h,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, 8345019);