#137154

order	512
nilpotency class	4
isoclinism	isoclinic class of #137153
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

677 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 677 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,7,8,5)(4,6)(9,10,15,21,40,20,39,19,16,23,38,51,14,22,13,37)(11,24,28,17,27,44,26,53,30,55,61,45,29,50,41,43)(12,31,34,64,33,18,46,48,36,52,66,47,35,49,32,65)(25,56,59,63,62,72,57,69,60,71,70,68,58,54,67,42);(1,2)(3,5)(4,6)(7,8)(9,11,14,27,16,30,40,29)(10,17,20,43,23,45,22,53)(12,25,35,62,36,60,33,58)(13,28,38,26,39,61,15,41)(18,42,52,68,49,69,31,63)(19,50,51,55,37,44,21,24)(32,59,66,57,46,70,34,67)(47,71,65,72,64,56,48,54);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,25)(13,32)(14,35)(15,34)(16,36)(17,42)(19,47)(20,52)(21,48)(22,31)(23,49)(24,54)(26,57)(27,62)(28,59)(29,58)(30,60)(33,40)(37,64)(38,66)(39,46)(41,67)(43,68)(44,56)(45,69)(50,71)(51,65)(53,63)(55,72)(61,70)

Polycyclic presentation:

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

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, 44350);