#8839307

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

4547 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)(10,12)(11,17)(16,20)(18,23)(25,26,30,33)(27,28,31,35)(29,32,38,40)(34,36,37,39);(3,7)(6,8)(9,10,14,12)(11,15,17,19)(13,20,21,16)(18,24,23,22)(25,27)(26,28)(29,37)(30,31)(32,39)(33,35)(34,38)(36,40);(1,2)(3,6)(4,5)(7,8)(11,17)(15,19)(18,23)(22,24)(27,31)(28,35)(34,37)(36,39);(9,11,14,17)(10,15,12,19)(13,18,21,23)(16,22,20,24)(25,28,30,35)(26,31,33,27)(29,36,38,39)(32,37,40,34);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,12,20)(11,18,17,23)(15,22,19,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,37)(33,40)(35,39)

Polycyclic presentation:

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

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, 8746503);