#409844

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2572
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

627 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)(3,7)(4,5)(6,8)(9,10,13,20,15,19,14,18)(11,16,23,30,22,31,28,32)(12,17,26,34,25,35,27,36)(21,37,24,33,38,29,39,40);(2,5)(6,7)(9,11,14,23,15,22,13,28)(10,16,20,32,19,31,18,30)(12,24,27,38,25,39,26,21)(17,33,34,37,35,40,36,29);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,21,22,38)(13,26)(14,27)(15,25)(16,29,31,37)(18,36)(19,35)(20,34)(23,24,28,39)(30,40,32,33);(10,18,19,20)(12,25)(16,30,31,32)(17,34,35,36)(21,38)(24,39)(26,27)(29,33,37,40)

Polycyclic presentation:

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

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, 317040);