#10363653

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

4322 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10270849);