#8958014

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

4319 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8865210);