#8125706

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

4601 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,f^2=g,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^c=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, 8032902);