#10181232

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	#43097	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4239 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:

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

Polycyclic presentation:

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

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, 10088428);