#8096673

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

4439 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:

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

Polycyclic presentation:

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

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, 8003869);