#494556

order	512
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1879
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

1279 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fhi,g^c=ghi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285805, #72285959, #72285988, …
maximal normal subgroup of	#72285805, #72285959, #72285988, …
maximal quotient of	#72321663, #72322904, #72342618

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401752);