#9923668

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

4608 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830864);