#411041

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

683 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,26,28,38)(13,19,29,18)(14,20)(15,30)(16,21)(22,33,23,37)(24,34)(25,36)(27,40,39,32)(31,35);(9,11,14,24,16,25,15,31)(10,17,20,34,21,36,30,35)(12,27,13,23,28,39,29,22)(18,37,26,32,19,33,38,40);(2,5)(6,8)(9,12,15,29,16,28,14,13)(10,18,20,26,21,19,30,38)(11,22,24,27,25,23,31,39)(17,32,35,37,36,40,34,33);(1,3,4,7)(2,6,5,8)(9,13,14,28,16,29,15,12)(10,19,20,38,21,18,30,26)(11,23,24,39,25,22,31,27)(17,33,34,40,36,37,35,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318237);