#93169

order	512
nilpotency class	4
isoclinism	isoclinic class of #93165
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#123	C₂²
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×#123 ▷ C₂⁴ ▷ C₂² ▷ C₁

523 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 48 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)(9,10,13,17)(11,19,21,33)(12,16,24,30)(14,18,26,32)(15,27,28,20)(22,34,35,40)(23,29,36,38)(25,31,37,39);(2,5)(9,11)(10,15)(12,23,25,20)(13,21,26,35)(14,22)(16,19,31,34)(17,28,32,38)(18,29)(24,36)(27,37)(30,33)(39,40)

Polycyclic presentation:

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

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, 365);