#415232

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
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

684 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)(9,10,15,20)(11,22,25,18)(12,17,30,37)(13,16,31,33)(14,21,23,19)(24,34,39,35)(26,40,27,32)(28,38,29,36);(1,3,5,4)(2,6,7,8)(9,11,23,31,15,25,14,13)(10,16,21,18,20,33,19,22)(12,26,28,24,30,27,29,39)(17,34,36,40,37,35,38,32);(1,4,5,3)(2,6,7,8)(9,12,14,28,15,30,23,29)(10,17,19,36,20,37,21,38)(11,24,13,27,25,39,31,26)(16,32,22,34,33,40,18,35);(9,13,14,25,15,31,23,11)(10,18,19,16,20,22,21,33)(12,27,28,39,30,26,29,24)(17,35,36,32,37,34,38,40)

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,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322428);