#93899

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	C₂⁴	#104
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

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

normal subgroups only

This group has 369 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2,3,5,4,6,7,8)(9,10,13,18)(11,21,23,51)(12,26,29,46)(14,20,34,47)(15,19,33,48)(16,36,37,60)(17,40,43,31)(22,39,54,45)(24,41,57,69)(25,52,56,66)(27,38,59,67)(28,42,61,65)(30,50,62,72)(32,63,35,58)(44,64,70,53)(49,68,71,55);(9,11,15,25)(10,16,19,38)(12,27,31,60)(13,32,33,57)(14,35,28,24)(17,41,26,58)(18,45,48,71)(20,49,42,39)(21,50,52,44)(22,53,55,30)(23,34,56,61)(29,54,40,68)(36,64,59,62)(37,65,67,47)(43,66,46,51)(63,70,69,72)

Polycyclic presentation:

2^9|a^2=d,b^2=i,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, 1095);