#129122

order	512
nilpotency class	3
isoclinism	isoclinic class of #108535
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₄⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1767
prime radical	C₂³	#423
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

252 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cg,f^a=fi,c^b=ch,d^b=dg,e^b=egi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36318);