#146731

order	512
nilpotency class	4
isoclinism	isoclinic class of #128583
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)(3,17,21,48)(4,23)(5,13)(6,30)(7,15)(8,14)(9,16)(10,19,37,31)(11,25)(12,28)(18,49,52,62)(20,38,56,45)(22,36,33,42)(24,40)(26,59)(27,43)(29,44)(32,46)(34,39)(35,54,61,53)(41,58)(47,55,63,51)(50,60,64,57);(1,3,7,22,9,21,8,33)(2,10,14,38,16,37,15,45)(4,18,26,53,27,52,34,54)(5,20,6,31,29,56,28,19)(11,35,40,49,41,61,46,62)(12,42,44,48,30,36,13,17)(23,57,39,47,43,60,59,63)(24,55,25,50,32,51,58,64);(1,4)(2,11)(3,18)(5,25)(6,32)(7,26)(8,34)(9,27)(10,35)(12,43)(13,39)(14,40)(15,46)(16,41)(17,47)(19,55)(20,50)(21,52)(22,53)(23,30)(24,28)(29,58)(31,51)(33,54)(36,57)(37,61)(38,49)(42,60)(44,59)(45,62)(48,63)(56,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=hi,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=dh,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285832, #72307180, #72307209, …
maximal normal subgroup of	#72285832, #72307180, #72307209, …
maximal quotient of	#72286000

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53927);