#107495

order	512
nilpotency class	3
isoclinism	isoclinic class of #107494
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

893 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)(11,21)(12,25)(13,19)(14,18)(15,20)(16,23)(17,27)(22,30)(24,34)(26,33)(28,32)(29,31)(35,36)(37,40)(38,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16,20,31)(12,22)(13,23)(14,29)(15,24)(17,30,33,40)(18,21,19,34)(25,35,32,39)(26,36)(27,38)(28,37);(3,8)(5,7)(9,12,15,28)(10,17,20,33)(11,22,24,37)(13,27,14,26)(16,30,31,40)(18,25,19,32)(21,35,34,39)(23,38,29,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,e^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,e^c=eh

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, 14691);