#493175

order	512
nilpotency class	2
isoclinism	isoclinic class of #180864
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

668 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)(3,6)(4,5)(7,8)(10,17,20,22)(12,29)(14,30)(18,32,36,23)(19,21,38,33)(24,34)(26,31)(35,39,40,37);(2,5)(6,8)(9,10,14,21,15,22,29,38,16,20,30,33,13,17,12,19)(11,23,26,35,27,36,24,39,28,32,31,40,25,18,34,37);(9,11)(10,18,20,36)(12,24)(13,25)(14,31)(15,27)(16,28)(17,32,22,23)(19,35,38,40)(21,39,33,37)(26,30)(29,34);(1,3,4,7)(2,6,5,8)(9,12,13,30,16,29,15,14)(10,19,17,33,20,38,22,21)(11,24,25,26,28,34,27,31)(18,35,32,37,36,40,23,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,d^a=di,g^a=gi,c^b=ch,f^b=fi,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237211, #72239377, #72239467, …
maximal normal subgroup of	#72237211, #72239377, #72239467, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400371);