#147455

order	512
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

764 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)(9,10)(11,19,21,36)(12,23)(13,16)(14,28)(15,20,30,39)(17,26)(18,27)(22,33,29,37)(24,34,40,31)(25,35)(32,38);(5,6,7,8)(9,11)(10,15)(12,24)(13,21)(14,29)(16,30)(17,33)(18,34)(19,35)(20,38)(22,27)(23,36)(25,40)(26,39)(28,31)(32,37);(1,2,3,4)(10,16)(12,25)(15,30)(18,28)(20,39)(24,40)(26,38)(31,34)

Polycyclic presentation:

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

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, 54651);