#138014

order	512
nilpotency class	4
isoclinism	isoclinic class of #137980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

377 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=ghi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,f^b=fh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45210);