#388469

order	512
nilpotency class	3
isoclinism	isoclinic class of #388466
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14520
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1139 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=eh,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318258, #72318894, #72318981
maximal normal subgroup of	#72318258, #72318894, #72318981

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295665);