#420734

order	512
nilpotency class	3
isoclinism	isoclinic class of #417704
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ei,d^2=e,f^2=i,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,f^c=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327930);