#94468

order	512
nilpotency class	5
isoclinism	isoclinic class of #94427
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1390
commutator	C₂²×Q₈	C₄²
Frattini, cosocle	C₄×C₄⋊Q₈	C₂²
prime radical	C₂³	#342
LCS	G ▷ C₂²×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₄×C₄⋊Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

208 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=eg,g^a=gi,h^a=hi,c^b=cg,d^b=dfi,g^b=gh,e^c=ehi,g^c=gi

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, 1664);