#494933

order	512
nilpotency class	4
isoclinism	isoclinic class of #13156
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1936
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1936
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

749 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)(3,16,22,46)(4,24,29,48)(5,12)(6,32)(7,34)(8,37)(9,15)(10,20,41,33)(11,28,45,36)(13,17)(14,31)(18,50)(19,47,23,40)(21,56,35,39)(25,42)(26,58,30,44)(27,59,51,43)(38,55)(49,57)(52,64)(53,62)(54,63)(60,61);(1,3,17,21,5,19,6,33,9,22,7,35,8,23,31,20)(2,10,32,40,37,56,13,46,15,41,14,47,12,39,34,16)(4,25,36,60,26,18,51,54,29,52,28,53,30,55,27,57)(11,42,48,63,59,38,58,61,45,64,24,49,43,50,44,62);(1,4,6,27,8,30,17,28,9,29,31,51,5,26,7,36)(2,11,13,44,12,43,32,24,15,45,34,58,37,59,14,48)(3,18,20,53,23,25,35,54,22,55,33,60,19,52,21,57)(10,38,16,49,39,42,47,61,41,50,46,63,56,64,40,62);(1,5,9,8)(2,12,15,37)(3,19,22,23)(4,26,29,30)(6,7,31,17)(10,39,41,56)(11,43,45,59)(13,32,34,14)(16,47,46,40)(18,52,55,25)(20,21,33,35)(24,58,48,44)(27,36,51,28)(38,42,50,64)(49,61,63,62)(53,57,60,54)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,d^2=i,e^2=h,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285749, #72285871, #72286096, …
maximal normal subgroup of	#72285749, #72285871, #72286096, …
maximal quotient of	#72310175, #72323046, #72342622

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402129);