#146314

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:

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

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,c^b=ch,d^b=di,g^b=gi,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239196, #72239198, #72239200
maximal normal subgroup of	#72239196, #72239198, #72239200

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53510);