#493134

order	512
nilpotency class	4
isoclinism	isoclinic class of #33404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#106
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×C₄×#106
prime radical	C₂⁴	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

713 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285601, #72285899, #72285959, …
maximal normal subgroup of	#72285601, #72285899, #72285959, …
maximal quotient of	#72310100, #72321765, M₁₆×#365

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400330);