#513207

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

Polycyclic presentation:

2^9|a^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,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	#72260483
maximal normal subgroup of	#72260483

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420403);