#128728

order	512
nilpotency class	3
isoclinism	isoclinic class of #106004
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1752
prime radical	C₂³	#435
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

267 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=gi,e^2=h,f^2=g,h^2=i,b^a=bd,d^a=dg,f^a=fg,c^b=cgh,d^b=di,e^b=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35924);