#107221

order	512
nilpotency class	3
isoclinism	isoclinic class of #107214
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

885 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14417);