#370592

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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

1014 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)(9,10,16,20,15,21,14,31)(11,22,12,18,25,33,24,17)(13,27,30,35,29,40,28,19)(23,34,26,39,38,32,37,36);(2,5)(6,7)(9,11,15,25)(10,17)(12,14,24,16)(13,23,29,38)(18,21)(19,32)(20,22)(26,28,37,30)(27,36)(31,33)(34,35)(39,40);(9,12,16,25,15,24,14,11)(10,18,31,22,21,17,20,33)(13,26,30,38,29,37,28,23)(19,34,40,36,35,32,27,39);(1,3,4,8)(2,6,5,7)(9,13,14,28,15,29,16,30)(10,19,20,27,21,35,31,40)(11,23,24,37,25,38,12,26)(17,32,22,36,18,34,33,39)

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

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, 277788);