#146769

order	512
nilpotency class	4
isoclinism	isoclinic class of #12750
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1786
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 ◁ C₂³×C₄ ◁ 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)(3,16,20,43)(4,23)(5,12)(6,32)(7,13)(8,15)(9,18,36,55)(10,25)(11,30)(14,33)(17,49,52,63)(19,37,29,45)(21,48,22,35)(24,46)(26,44)(27,38)(28,41)(31,42)(34,53,62,57)(39,60)(40,59)(47,54,64,51)(50,61,56,58);(1,3,7,21,8,20,33,22)(2,9,13,37,15,36,14,45)(4,17,26,53,28,52,27,57)(5,29,30,55,31,19,6,18)(10,34,39,49,40,62,46,63)(11,35,42,16,32,48,12,43)(23,58,44,64,41,61,38,47)(24,54,25,50,60,51,59,56);(1,4)(2,10)(3,17)(5,25)(6,24)(7,26)(8,28)(9,34)(11,41)(12,44)(13,39)(14,46)(15,40)(16,47)(18,54)(19,56)(20,52)(21,53)(22,57)(23,32)(27,33)(29,50)(30,60)(31,59)(35,61)(36,62)(37,49)(38,42)(43,64)(45,63)(48,58)(51,55)

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,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=eh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307178, #72307184, #72307258, …
maximal normal subgroup of	#72307178, #72307184, #72307258, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53965);