#8924365

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

4502 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)(10,15)(11,17)(13,21)(20,24)(25,26,30,33)(27,34,29,31)(28,32,37,40)(35,39,36,38);(3,7)(6,8)(9,10,14,15)(11,16,17,12)(13,18,21,23)(19,24,22,20)(27,29)(31,34)(35,36)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,17)(10,16,15,12)(13,22,21,19)(18,20,23,24)(25,27,30,29)(26,31,33,34)(28,36,37,35)(32,39,40,38);(9,11)(10,16)(12,15)(13,19)(14,17)(18,24)(20,23)(21,22)(25,28,30,37)(26,32,33,40)(27,35,29,36)(31,38,34,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,16)(10,17,15,11)(13,20,21,24)(18,22,23,19)(25,29,30,27)(26,31,33,34)(28,35,37,36)(32,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,18,15,23)(11,19,17,22)(12,20,16,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,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, 8831561);