#8475121

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

4505 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)(4,6)(5,7)(12,20)(16,21)(18,24)(22,23)(25,26)(27,32)(28,34)(29,33)(30,35)(31,36)(37,38)(39,40);(1,3,6,5)(2,7,4,8)(9,10)(11,15)(12,21)(13,17)(14,19)(16,20)(18,22)(23,24)(25,27,31,29)(26,32,36,33)(28,39,35,37)(30,38,34,40);(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,24)(21,23)(25,27,31,29)(26,33,36,32)(28,37,35,39)(30,38,34,40);(1,2,6,4)(3,7,5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,24)(21,22)(25,28,31,35)(26,34,36,30)(27,37,29,39)(32,38,33,40);(1,4,6,2)(3,7,5,8)(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,21)(19,24)(25,29,31,27)(26,32,36,33)(28,39,35,37)(30,40,34,38);(1,5,6,3)(2,7,4,8)(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,30,31,34)(26,35,36,28)(27,38,29,40)(32,37,33,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gh,e^2=gh,f^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, 8382317);