#146655

order	512
nilpotency class	4
isoclinism	isoclinic class of #12753
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1790
commutator	C₂×D₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10745
prime radical	#3449	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

960 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282255, #72317352, #72317357, …
maximal normal subgroup of	#72282255, #72317352, #72317357, …
maximal quotient of	C₂×#146655

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53851);