#8926581

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

4356 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)(9,10,14,16)(11,15,17,12)(13,18,22,23)(19,21,24,20)(25,26)(27,34)(28,33)(29,30)(31,35)(32,38)(36,40)(37,39);(2,3)(6,7)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,21,23,20)(25,27,31,38)(26,32,35,34)(28,36,39,29)(30,37,40,33);(1,3,5,2)(4,7,8,6)(10,16)(11,17)(18,23)(19,24)(25,28,31,39)(26,33,35,37)(27,36,38,29)(30,34,40,32);(1,4)(2,6)(3,7)(5,8)(9,12,14,15)(10,17,16,11)(13,21,22,20)(18,19,23,24)(25,29)(26,30)(27,28)(31,36)(32,37)(33,34)(35,40)(38,39);(9,13,14,22)(10,18,16,23)(11,19,17,24)(12,20,15,21)(25,27)(26,34)(28,36)(29,39)(30,37)(31,38)(32,35)(33,40);(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,24)(22,23)(25,30)(26,29)(27,37)(28,32)(31,40)(33,38)(34,39)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833777);