#146741

order	512
nilpotency class	4
isoclinism	isoclinic class of #128584
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10788
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

943 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dhi,e^b=ehi,h^b=hi,d^c=dhi,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307180, #72307193, #72307268, …
maximal normal subgroup of	#72307180, #72307193, #72307268, …
maximal quotient of	#72318980

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53937);