#370610

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1018 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)(4,5)(6,8)(9,10)(11,22)(12,26,31,17)(13,16)(14,20)(15,21)(18,25)(19,23)(24,36,39,35)(27,33,28,40)(29,37,30,38)(32,34);(2,5)(6,7)(9,11,23,32,15,25,14,13)(10,16,20,18,21,34,19,22)(12,27,29,24,31,28,30,39)(17,35,37,40,26,36,38,33);(9,12,23,30,15,31,14,29)(10,17,20,38,21,26,19,37)(11,24,32,27,25,39,13,28)(16,33,18,36,34,40,22,35);(1,3,4,8)(2,6,5,7)(9,13,14,25,15,32,23,11)(10,18,19,16,21,22,20,34)(12,28,29,39,31,27,30,24)(17,36,37,33,26,35,38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=fg,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277806);