#8424553

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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8331749);