#10107013

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	#59407	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4405 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=gh,e^2=g,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, 10014209);