#370593

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
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

1004 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)(9,10,16,20,15,21,14,32)(11,22,12,18,25,34,24,17)(13,27,31,36,30,40,29,19)(23,35,26,39,38,33,28,37);(2,5)(6,7)(9,11,15,25)(10,17)(12,14,24,16)(13,23,30,38)(18,21)(19,33)(20,22)(26,29,28,31)(27,37)(32,34)(35,36)(39,40);(9,12,16,25,15,24,14,11)(10,18,32,22,21,17,20,34)(13,28,31,23,30,26,29,38)(19,33,40,39,36,35,27,37);(1,3,4,8)(2,6,5,7)(9,13,14,29,15,30,16,31)(10,19,20,27,21,36,32,40)(11,23,24,28,25,38,12,26)(17,33,22,37,18,35,34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=fg,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,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, 277789);