#417806

order	512
nilpotency class	3
isoclinism	isoclinic class of #417805
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15258
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264906, #72265041, #72341340, …
maximal normal subgroup of	#72264906, #72265041, #72341340, …
maximal quotient of	#72265207

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325002);