#93161

order	512
nilpotency class	4
isoclinism	isoclinic class of #93155
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

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	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×#123 ▷ C₂⁴ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 531 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)(3,6,8,5)(4,7)(9,10,13,17,14,18,27,34)(11,19,21,35,22,36,26,39)(12,16,24,31,25,32,38,40)(15,28,29,20,30,37,33,23);(2,5,7,6)(9,11)(10,15,18,30)(12,23,25,20)(13,26,27,21)(14,22)(16,19)(17,33)(24,37)(28,38)(29,34)(31,39,40,35)(32,36)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=i,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=gi,c^b=cg,d^b=dfh,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, 357);