#93898

order	512
nilpotency class	5
isoclinism	isoclinic class of #7398
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂²×C₈	C₂×C₈
Frattini, cosocle	#1729	C₂²
prime radical	C₂³×D₈	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ #1729 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 433 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=fi,d^2=g,f^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=def,f^b=fi,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, 1094);