#129790

order	512
nilpotency class	4
isoclinism	isoclinic class of #129789
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#60459	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

860 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 108 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)(11,23,27,51)(12,29,32,45)(13,34)(14,20)(15,21)(16,22)(17,25,41,57)(18,31,44,33)(19,36)(24,52)(26,50,28,46)(30,61,64,43)(35,42,58,40)(37,48)(38,47)(39,66)(49,55)(53,67)(54,71)(56,70)(59,72)(60,63,69,65)(62,68);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,24)(13,35)(14,26)(15,28)(16,27)(18,39)(19,46)(20,40)(21,42)(22,41)(23,47)(25,37)(29,59)(30,62)(31,54)(32,56)(33,55)(34,51)(36,57)(38,58)(43,52)(44,68)(45,67)(48,50)(49,69)(53,63)(60,71)(61,70)(64,66)(65,72);(9,12,16,32)(10,18,22,44)(11,24,27,56)(13,30,38,64)(14,33,15,31)(17,39,41,68)(19,43,48,61)(20,45,21,29)(23,49,51,71)(25,53,57,72)(26,55,28,54)(34,60,47,69)(35,62,58,66)(36,65,37,63)(40,67,42,59)(46,52,50,70)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36986);