#8449365

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

4558 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8356561);