#10361121

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

4643 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)(11,18)(12,13)(14,15)(16,19)(17,22)(20,24)(21,23)(25,26,30,32)(27,33,35,29)(28,31,36,39)(34,40,37,38);(1,2)(3,7)(4,6)(5,8)(10,15)(11,19)(13,22)(20,23)(25,27,30,35)(26,29,32,33)(28,34,36,37)(31,38,39,40);(1,2,4,6)(3,5,8,7)(10,15)(12,17)(16,18)(20,23)(28,36)(31,39)(34,37)(38,40);(1,2,4,6)(3,5,8,7)(9,11,14,19)(10,16,15,18)(12,20,17,23)(13,21,22,24)(25,27)(26,29)(28,37)(30,35)(31,40)(32,33)(34,36)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,17)(10,13,15,22)(11,20,19,23)(16,21,18,24)(25,28,30,36)(26,31,32,39)(27,34,35,37)(29,38,33,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,17)(11,21)(12,15)(14,22)(16,23)(18,20)(19,24)(25,29)(26,27)(28,38)(30,33)(31,34)(32,35)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=hi,e^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, 10268317);