#146653

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

962 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)(3,14)(4,19)(5,23,25,11)(6,26)(7,12)(8,13)(9,16,32,49)(10,21)(15,43)(17,41)(18,42)(20,53,55,52)(22,34)(24,37,29,38)(27,40)(28,35)(30,58,60,64)(31,50,57,48)(33,56,36,54)(39,45)(44,61,63,51)(46,59)(47,62);(1,3,7,17,8,18,6,27)(2,9,12,31,13,32,26,57)(4,15,21,46,22,47,28,45)(5,16,24,50,25,49,29,48)(10,30,34,61,35,60,19,51)(11,14,38,41,23,42,37,40)(20,44,56,58,55,63,54,64)(33,59,53,62,36,39,52,43);(1,4)(2,10)(3,15)(5,20)(6,28)(7,21)(8,22)(9,30)(11,36)(12,34)(13,35)(14,39)(16,44)(17,46)(18,47)(19,26)(23,33)(24,56)(25,55)(27,45)(29,54)(31,61)(32,60)(37,53)(38,52)(40,62)(41,43)(42,59)(48,64)(49,63)(50,58)(51,57)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^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	#72282473, #72317344, #72317366, …
maximal normal subgroup of	#72282473, #72317344, #72317366, …
maximal quotient of	C₂×#146653

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53849);