#140596

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	#3150	C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

392 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cgh,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47792);