#409578

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

626 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)(4,6)(5,8)(9,10,14,18,15,19,13,20)(11,21,23,32,25,16,24,33)(12,26,29,35,28,36,30,17)(22,34,27,37,38,40,39,31);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,32)(19,21)(20,33)(22,29)(26,31)(28,39)(30,38)(35,40)(36,37);(1,3,5,4)(2,7,8,6)(9,12,13,29,15,28,14,30)(10,17,18,36,19,35,20,26)(11,22,23,39,25,38,24,27)(16,31,32,34,21,37,33,40);(1,4,5,3)(2,6,8,7)(10,18,19,20)(12,28)(16,32,21,33)(17,26,35,36)(22,38)(27,39)(29,30)(31,40,37,34)

Polycyclic presentation:

2^9|a^2=g,b^2=efh,c^2=ef,d^2=e,f^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,g^b=gi,d^c=di,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, 316774);