#495237

order	512
nilpotency class	4
isoclinism	isoclinic class of #13247
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1972
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1972
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1047 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)(4,23,28,54)(5,12)(6,31)(7,13)(8,34)(9,15)(10,19)(11,27,42,56)(14,32)(17,35,52,62)(18,37)(20,36)(21,44)(22,47)(24,55,49,39)(25,41,29,60)(26,58,48,40)(30,43)(33,46)(38,53)(45,51,63,57)(50,61,59,64);(1,3,7,33,8,21,32,53,9,22,30,20,5,18,6,19)(2,10,31,37,12,36,43,47,15,38,14,44,34,46,13,16)(4,24,56,50,29,52,26,57,28,49,27,59,25,17,48,51)(11,39,23,45,40,62,41,61,42,55,54,63,58,35,60,64);(1,4,7,27,8,29,32,48,9,28,30,56,5,25,6,26)(2,11,14,23,12,40,13,41,15,42,31,54,34,58,43,60)(3,17,20,51,21,24,19,50,22,52,33,57,18,49,53,59)(10,35,37,61,36,39,47,63,38,62,44,64,46,55,16,45);(1,5,9,8)(2,12,15,34)(3,18,22,21)(4,25,28,29)(6,30,32,7)(10,36,38,46)(11,40,42,58)(13,31,43,14)(16,37,47,44)(17,49,52,24)(19,20,53,33)(23,41,54,60)(26,56,48,27)(35,39,62,55)(45,61,63,64)(50,51,59,57)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285749, #72285959, #72286016, …
maximal normal subgroup of	#72285749, #72285959, #72286016, …
maximal quotient of	#72322895, #72342623

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402433);