#493173

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
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×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

708 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)(9,10)(11,23,26,35)(12,18)(13,19)(14,31)(15,21)(16,22)(17,25,33,28)(20,29)(24,34,39,40)(27,32,38,37)(30,36);(2,5)(6,8)(9,11,14,27,15,28,30,39,16,26,29,38,13,25,12,24)(10,17,20,34,21,35,18,32,22,33,31,40,19,23,36,37);(9,10)(11,23,26,35)(12,18)(13,19)(14,31)(15,21)(16,22)(17,25,33,28)(20,29)(24,34,39,40)(27,32,38,37)(30,36);(1,3,4,7)(2,6,5,8)(9,12,13,29,16,30,15,14)(10,18,19,20,22,36,21,31)(11,24,25,38,26,39,28,27)(17,32,23,34,33,37,35,40)

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,g^a=gi,c^b=ch,f^b=fi,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237205, #72237209, #72237211, …
maximal normal subgroup of	#72237205, #72237209, #72237211, …
maximal quotient of	C₂×#493173

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400369);