#8467052

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

4599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=h,e^2=ghi,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, 8374248);