#494512

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1879
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

1283 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307178, #72307198, #72307258, …
maximal normal subgroup of	#72307178, #72307198, #72307258, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401708);