#8932817

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
prime radical	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4437 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)(11,17)(12,20)(15,18)(16,21)(25,26)(27,32)(28,37)(29,38)(30,40)(31,33)(34,36)(35,39);(2,3)(6,7)(9,10)(11,18)(12,21)(13,14)(15,17)(16,20)(19,23)(22,24)(25,27,31,35)(26,32,33,39)(28,38,40,34)(29,30,36,37);(1,3,5,2)(4,7,8,6)(10,14)(15,18)(16,21)(23,24)(26,33)(27,35)(30,37)(34,38);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,22)(13,17)(14,18)(16,24)(19,20)(21,23)(25,28)(26,30)(27,34)(29,32)(31,40)(33,37)(35,38)(36,39);(9,12)(10,16)(11,19)(13,20)(14,21)(15,23)(17,22)(18,24)(25,29,31,36)(26,34,33,38)(27,30,35,37)(28,39,40,32);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30)(26,28)(27,36)(29,35)(31,37)(32,38)(33,40)(34,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840013);