#10096972

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

4427 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=gh,e^2=g,f^2=gh,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, 10004168);