#513232

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

367 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=i,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=de,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234374, #72322373, #72322387
maximal normal subgroup of	#72234374, #72322373, #72322387

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420428);