#512662

order	512
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#127	C₂⁴
socle	C₂	C₂×#2127
prime radical	#60797	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3449 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

818 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)(3,14,17,8)(4,9)(5,10)(6,12,26,11)(7,13)(15,38,41,27)(16,39,42,28)(18,29,25,30)(19,44,46,53)(20,31)(21,33)(22,32,49,37)(23,35,51,34)(24,36)(40,59,60,52)(43,54,50,55)(45,61,62,64)(47,57)(48,56,63,58);(1,3)(2,8)(4,19)(5,16)(6,25)(7,18)(9,27)(10,28)(11,29)(12,14)(13,30)(15,21)(17,26)(20,45)(22,41)(23,50)(24,43)(31,52)(32,38)(33,53)(34,54)(35,39)(36,55)(37,44)(40,47)(42,51)(46,49)(48,60)(56,59)(57,64)(58,61)(62,63);(1,4)(2,9)(3,15)(5,20)(6,22)(7,21)(8,27)(10,31)(11,37)(12,32)(13,33)(14,38)(16,40)(17,41)(18,19)(23,48)(24,47)(25,46)(26,49)(28,52)(29,44)(30,53)(34,58)(35,56)(36,57)(39,59)(42,60)(43,45)(50,62)(51,63)(54,61)(55,64);(1,5,7,24)(2,10,13,36)(3,16,18,43)(4,20,21,47)(6,23,26,51)(8,28,30,55)(9,31,33,57)(11,34,12,35)(14,39,29,54)(15,40,19,45)(17,42,25,50)(22,48,49,63)(27,52,53,64)(32,56,37,58)(38,59,44,61)(41,60,46,62)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=hi,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ehi,f^b=fgh,g^b=gi,e^c=ei,f^e=fhi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284440, #72318931, #72318941
maximal normal subgroup of	#72284440, #72318931, #72318941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419858);