#146313

order	512
nilpotency class	3
isoclinism	isoclinic class of #10792
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10694
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

553 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=i,e^2=h,h^2=i,b^a=bd,d^a=di,f^a=fi,c^b=ch,g^b=gi,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239196, #72239199, #72239200
maximal normal subgroup of	#72239196, #72239199, #72239200

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53509);