#422474

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=ei,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329670);