#8458096

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

4491 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=gh,e^2=ghi,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, 8365292);