#512648

order	512
nilpotency class	5
isoclinism	isoclinic class of #512647
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	#60807	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

802 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)(3,13,17,40)(4,9)(5,21,23,55)(6,25,27,11)(7,12)(8,16,30,47)(10,24)(14,39,44,61)(15,36,26,41)(18,49,22,54)(19,51,53,34)(20,35)(28,46,59,64)(29,45,31,48)(32,43,60,63)(33,50)(37,56)(38,57)(42,58,52,62);(1,3)(2,8)(4,14)(5,16)(6,26)(7,15)(9,32)(10,36)(11,30)(12,29)(13,38)(17,27)(18,43)(19,52)(20,42)(21,41)(22,46)(23,45)(24,47)(25,31)(28,35)(33,58)(34,60)(37,39)(40,55)(44,53)(48,57)(49,62)(50,63)(51,59)(54,61)(56,64);(1,4,7,20)(2,9,12,35)(3,14,15,42)(5,22,23,18)(6,19,27,53)(8,28,29,32)(10,37,38,33)(11,34,25,51)(13,39,36,58)(16,46,45,43)(17,44,26,52)(21,54,55,49)(24,56,57,50)(30,59,31,60)(40,61,41,62)(47,64,48,63);(3,15)(8,29)(13,36)(14,42)(16,45)(17,26)(28,32)(30,31)(39,58)(40,41)(43,46)(44,52)(47,48)(59,60)(61,62)(63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284730, #72286802
maximal normal subgroup of	#72284730, #72286802

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419844);