#413146

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1044 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ef,c^2=e,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 320342);