#493162

order	512
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×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×#1776
prime radical	C₂⁴	#137
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

687 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285705, #72285899, #72286027, …
maximal normal subgroup of	#72285705, #72285899, #72286027, …
maximal quotient of	#72310155, #72321883, #72342549

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400358);