#106554

order	512
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 13750);