#373445

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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×C₄⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

932 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)(4,6)(5,8)(9,10,15,19)(11,12)(13,28)(14,18,21,20)(16,32)(17,31)(22,33,37,40)(23,26)(24,27)(25,36,39,38)(29,34)(30,35);(1,3,5,4)(2,6,8,7)(9,11,21,23,15,24,14,32)(10,16,20,12,19,26,18,27)(13,22,29,38,30,37,31,36)(17,33,28,25,34,40,35,39);(1,3,5,4)(2,7,8,6)(9,12,14,16,15,27,21,26)(10,11,20,23,19,24,18,32)(13,25,31,33,30,39,29,40)(17,22,28,38,34,37,35,36);(1,4,5,3)(2,7,8,6)(9,13,15,30)(10,17,19,34)(11,22,24,37)(12,25,27,39)(14,31,21,29)(16,33,26,40)(18,35,20,28)(23,38,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280641);