#368177

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1444 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)(9,10,13,18,14,17,25,32)(11,15,21,28,20,27,35,38)(12,22,24,36,23,16,29,31)(19,26,34,37,33,30,39,40);(2,5)(6,8)(9,11,13,21,14,20,25,35)(10,15,18,28,17,27,32,38)(12,19,24,34,23,33,29,39)(16,30,31,40,22,26,36,37);(1,3,4,7)(2,6,5,8)(9,12)(10,16,17,22)(11,19)(13,24)(14,23)(15,26,27,30)(18,31,32,36)(20,33)(21,34)(25,29)(28,37,38,40)(35,39);(2,5)(6,8)(12,23)(16,22)(19,33)(24,29)(26,30)(31,36)(34,39)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275373);