#373382

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

776 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)(4,8)(5,6)(9,10,16,20,15,21,14,22)(11,23,26,17,27,37,28,36)(12,18,30,39,31,25,34,38)(13,24,32,19,33,40,35,29);(2,6)(7,8)(9,11,16,26,15,27,14,28)(10,17,20,37,21,36,22,23)(12,29,30,24,31,19,34,40)(13,25,32,38,33,18,35,39);(1,3,5,4)(2,7,6,8)(9,12,16,30,15,31,14,34)(10,18,22,38,21,25,20,39)(11,24,28,29,27,40,26,19)(13,23,32,17,33,37,35,36);(1,4,5,3)(2,7,6,8)(9,13,15,33)(10,19,21,29)(11,25,27,18)(12,23,31,37)(14,35,16,32)(17,34,36,30)(20,40,22,24)(26,38,28,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ef,d^2=efg,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,e^b=ei,g^b=gi,e^c=ei,f^c=fi,e^d=ei,f^d=fi

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, 280578);