#10121235

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

4497 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^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, 10028431);