#8463953

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

4428 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:

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

Polycyclic presentation:

2^9|c^2=gi,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, 8371149);