#8106007

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

4625 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013203);