#106669

order	512
nilpotency class	3
isoclinism	isoclinic class of #106654
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

803 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,f^b=fh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13865);