#415233

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
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

674 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:

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

Polycyclic presentation:

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

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, 322429);