#8472839

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

4599 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

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, 8380035);