#9028042

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8935238);