#8552113

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

4503 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8459309);