#93053

order	512
nilpotency class	4
isoclinism	isoclinic class of #93047
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1294
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂²×C₄⋊C₈	C₂²
prime radical	C₂⁴	#103
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

282 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 128

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=h,e^2=i,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,c^b=cg,d^b=dh,f^b=fh,g^b=gi,e^c=ehi

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, 249);