#133601

order	512
nilpotency class	4
isoclinism	isoclinic class of #133338
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂³	#379
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

485 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18)(19,29,32,44)(20,33)(21,26)(22,37)(23,28)(24,30,42,35)(25,34)(27,39)(31,40,38,45)(36,43)(41,47,48,46);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,24)(20,30)(21,35)(22,38)(23,32)(25,40)(26,44)(27,41)(28,42)(29,33)(31,39)(34,47)(36,46)(37,48)(43,45);(17,20,23,21)(18,25,28,43)(19,30,32,35)(22,36,39,34)(24,40,42,45)(26,37,33,27)(29,41,44,48)(31,47,38,46)

Polycyclic presentation:

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

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, 40797);