#133759

order	512
nilpotency class	4
isoclinism	isoclinic class of #133290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 743 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) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40955);