#9041132

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	#47992	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4333 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,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, 8948328);