#147033

order	512
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

630 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,d^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dh,f^b=fi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54229);