#8174391

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

4410 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)(3,7)(10,15)(11,19)(13,17)(20,22)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(1,2,4,5)(3,6,7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(11,19)(16,23)(20,22)(21,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,16,15,23)(12,22,18,20)(13,21,17,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,18)(10,17,15,13)(11,20,19,22)(16,21,23,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(9,13,14,17)(10,18,15,12)(11,21,19,24)(16,20,23,22)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

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, 8081587);