#493088

order	512
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#442
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1762
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1095 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285775, #72285899, #72285988, …
maximal normal subgroup of	#72285775, #72285899, #72285988, …
maximal quotient of	#72321464, #72342576

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400284);