#93082

order	512
nilpotency class	4
isoclinism	isoclinic class of #93073
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1296
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
prime radical	C₂³	#332
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

146 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 256

A minimal faithful permutation representation — 2 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278);