#138288

order	512
nilpotency class	4
isoclinism	isoclinic class of #138132
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

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	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

425 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=ghi,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=ei,f^c=fi

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, 45484);