#386142

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
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

923 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:

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

Polycyclic presentation:

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

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, 293338);