#513197

order	512
nilpotency class	5
isoclinism	isoclinic class of D₄×D₃₂
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
prime radical	#33846	C₂
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

364 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)(11,16)(12,17)(13,30)(14,33)(15,21)(18,26)(19,27)(20,31)(22,34)(23,50)(24,53)(25,38)(28,45)(29,41)(32,59)(35,46)(36,47)(37,51)(39,55)(40,54)(42,58)(43,56)(44,57)(48,66)(49,62)(52,72)(60,68)(61,67)(63,71)(64,69)(65,70);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,21,38)(12,22,29,49)(13,23,32,52)(14,24,31,51)(17,34,41,62)(18,35,43,64)(19,36,44,65)(20,37,33,53)(26,46,56,69)(27,47,57,70)(28,48,54,67)(30,50,59,72)(39,60,42,63)(40,61,45,66)(55,68,58,71);(10,17)(12,26)(13,27)(14,31)(16,34)(18,19)(20,45)(21,41)(22,46)(23,47)(24,51)(28,58)(29,56)(30,39)(32,57)(33,40)(35,36)(37,66)(38,62)(42,59)(43,44)(48,71)(49,69)(50,60)(52,70)(53,61)(54,55)(63,72)(64,65)(67,68);(3,8)(5,7)

Polycyclic presentation:

2^9|a^2=e,b^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,d^b=de,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234374, #72322380, #72322434
maximal normal subgroup of	#72234374, #72322380, #72322434

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420393);