#8614108

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

4566 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8521304);