#9054402

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	#35470	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=g,e^2=ghi,f^2=h,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, 8961598);