#93894

order	512
nilpotency class	5
isoclinism	isoclinic class of #93701
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1316
commutator	C₂³×C₄	C₂×C₈
Frattini, cosocle	#1728	C₂²
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1728 ◁ G
LECS	G ▷ #1728 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 385 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=f,d^2=g,g^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=def,g^b=gi,d^c=dh,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1090);