#500162

order	512
nilpotency class	4
isoclinism	isoclinic class of #500160
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24033
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61090	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

950 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)(3,15,18,45)(4,10,24,41)(5,25,29,12)(6,30,26,11)(7,31,28,13)(9,33,36,19)(16,37,48,43)(17,49,50,44)(20,34,32,46)(21,39,27,59)(22,38,56,42)(23,57,55,40)(35,53,61,51)(47,64,54,62)(52,63,58,60);(3,16)(5,26)(6,29)(7,28)(9,34)(11,25)(12,30)(13,31)(15,43)(17,47)(18,48)(19,32)(20,33)(21,22)(23,55)(27,56)(35,60)(36,46)(37,45)(38,39)(40,57)(42,59)(44,64)(49,62)(50,54)(51,52)(53,58)(61,63);(1,3,7,32,8,18,28,20)(2,9,13,43,14,36,31,37)(4,17,23,58,24,50,55,52)(5,16,26,33,29,48,6,19)(10,35,40,64,41,61,57,62)(11,34,25,45,30,46,12,15)(21,54,22,53,27,47,56,51)(38,63,39,44,42,60,59,49);(1,4)(2,10)(3,17)(5,27)(6,22)(7,23)(8,24)(9,35)(11,42)(12,39)(13,40)(14,41)(15,44)(16,47)(18,50)(19,53)(20,52)(21,29)(25,59)(26,56)(28,55)(30,38)(31,57)(32,58)(33,51)(34,60)(36,61)(37,62)(43,64)(45,49)(46,63)(48,54)

Polycyclic presentation:

2^9|a^2=i,c^2=eh,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,e^c=ei,f^c=fhi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250052, #72250066, #72250223, …
maximal normal subgroup of	#72250052, #72250066, #72250223, …
maximal quotient of	#72250533, #72281975

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407358);