#494329

order	512
nilpotency class	4
isoclinism	isoclinic class of #12952
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1864
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1025 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)(3,16,20,50)(4,21,25,42)(5,12)(6,28)(7,31)(8,34)(9,15)(10,19,39,56)(11,24,43,23)(13,27)(14,26)(17,37)(18,47,35,45)(22,48,36,46)(29,40,32,38)(30,44,33,41)(49,58)(51,59)(52,63)(53,62)(54,61)(55,64)(57,60);(1,3,8,35,9,20,5,18)(2,10,12,38,15,39,34,40)(4,17,36,53,25,54,22,51)(6,29,27,19,7,32,26,56)(11,37,41,59,43,61,44,62)(13,45,28,16,14,47,31,50)(21,49,48,60,42,63,46,64)(23,57,30,58,24,55,33,52);(1,4)(2,11)(3,17)(5,22)(6,30)(7,33)(8,36)(9,25)(10,37)(12,41)(13,46)(14,48)(15,43)(16,49)(18,51)(19,55)(20,54)(21,28)(23,26)(24,27)(29,58)(31,42)(32,52)(34,44)(35,53)(38,59)(39,61)(40,62)(45,64)(47,60)(50,63)(56,57);(1,5,9,8)(2,12,15,34)(3,18,20,35)(4,22,25,36)(6,26,7,27)(10,38,39,40)(11,41,43,44)(13,28,14,31)(16,47,50,45)(17,51,54,53)(19,29,56,32)(21,48,42,46)(23,33,24,30)(37,59,61,62)(49,60,63,64)(52,55,58,57)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285805, #72285871, #72286027, …
maximal normal subgroup of	#72285805, #72285871, #72286027, …
maximal quotient of	#72310164, #72321645, #72322861, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401525);