#136428

order	512
nilpotency class	4
isoclinism	isoclinic class of #136369
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1853
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

357 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,d^2=g,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ehi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43624);