#146298

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₄²
commutator	C₄	C₄²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=i,e^2=h,h^2=i,b^a=bd,f^a=fi,c^b=ci,e^b=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53494);