#107340

order	512
nilpotency class	3
isoclinism	isoclinic class of #107334
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

892 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=ehi,f^b=fi,d^c=dhi,e^c=eh,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, 14536);