#9040458

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	#47995	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4515 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)(3,4)(5,6)(7,8)(9,10,13,15)(11,14,17,22)(12,16,20,23)(18,21,19,24);(1,2,5,6)(3,4,7,8)(9,11)(10,14)(12,19)(13,17)(15,22)(16,24)(18,20)(21,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(1,3,5,7)(2,4,6,8)(10,15)(11,17)(16,23)(18,19)(26,30)(28,37)(31,33)(34,38);(2,6)(4,8)(9,12,13,20)(10,16,15,23)(11,18,17,19)(14,21,22,24)(25,27)(26,31)(28,38)(29,35)(30,33)(32,40)(34,37)(36,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,19)(13,17)(15,22)(16,24)(18,20)(21,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,24)(20,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,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, 8947654);