#111564

order	512
nilpotency class	3
isoclinism	isoclinic class of #2008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1089 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,d^c=dh,e^c=eh,g^c=gh

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, 18760);