#10181076

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	#43822	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4213 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=ghi,e^2=gi,f^2=g,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, 10088272);