#8933752

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	#47031	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4578 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=h,f^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,d^c=dgi,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, 8840948);