#94869

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	#95
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=g,e^2=h,f^2=i,b^a=bd,f^a=fg,c^b=ch,e^b=eg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2065);