#10220371

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

4193 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10127567);