#9040969

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	#35355	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4614 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=g,e^2=g,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, 8948165);