#500161

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:

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

Polycyclic presentation:

2^9|b^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	#72307079, #72307089, #72307224, …
maximal normal subgroup of	#72307079, #72307089, #72307224, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407357);