#477249

order	512
nilpotency class	3
isoclinism	isoclinic class of #476106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

862 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,h^c=hi,e^d=ei

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, 384445);