#133716

order	512
nilpotency class	4
isoclinism	isoclinic class of #133341
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

525 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

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, 40912);