#8932164

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

4368 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)(4,7,8,6)(9,10,14,15)(11,12,17,16)(13,21,22,18)(19,23,24,20)(25,26)(27,29)(28,36)(30,34)(31,33)(32,38)(35,39)(37,40);(2,3)(6,7)(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,23)(22,24)(25,26,30,34)(27,29,33,31)(28,32,38,36)(35,37,40,39);(1,3,5,2)(4,7,8,6)(10,15)(11,17)(13,22)(20,23)(25,27,30,33)(26,31,34,29)(28,35,38,40)(32,39,36,37);(1,4)(2,6)(3,7)(5,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,38)(37,40);(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,40,36,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839360);