#149476

order	512
nilpotency class	4
isoclinism	isoclinic class of #129627
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	C₂⁴	#120
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

781 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 781 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=eh,g^b=ghi,d^c=dh,g^c=gi,e^d=ei,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56672);