#133346

order	512
nilpotency class	4
isoclinism	isoclinic class of #133338
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

802 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,7,8,5)(4,6)(9,10,15,21,16,22,14,20)(11,23,25,39,26,37,31,33)(12,27,29,18,13,19,28,36)(17,32,34,24,35,40,38,30);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,24)(13,30)(14,31)(15,25)(16,26)(18,33)(19,37)(20,34)(21,38)(22,35)(23,27)(28,32)(29,40)(36,39);(9,12,16,13)(10,18,22,36)(11,24,26,30)(14,29,15,28)(17,33,35,39)(19,21,27,20)(23,34,37,38)(25,32,31,40)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40542);