#392713

order	512
nilpotency class	3
isoclinism	isoclinic class of #392712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14694
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3634 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)(3,9,17,33)(4,20,22,37)(5,24,25,38)(6,13)(7,14)(8,12)(10,23,35,27)(11,26,36,28)(15,39)(16,43)(18,31,19,32)(21,34)(29,46)(30,47)(40,55,60,64)(41,56)(42,48)(44,57)(45,53)(49,50)(51,54)(52,58)(59,62,63,61);(1,3)(2,9,12,33)(4,15)(5,16)(6,19)(7,18)(8,17)(10,29,35,56)(11,30,36,57)(13,32,14,31)(20,48,37,39)(21,40)(22,42)(23,46)(24,53,38,43)(25,45)(26,47)(27,41)(28,44)(34,55,58,64)(49,63,54,59)(50,62)(51,61)(52,60);(1,4,8,22)(2,10,12,35)(3,15,17,42)(5,21,25,52)(6,27,7,23)(9,29,33,56)(11,34,36,58)(13,37,14,20)(16,40,45,60)(18,46,19,41)(24,54,38,49)(26,51,28,50)(30,55,57,64)(31,48,32,39)(43,63,53,59)(44,62,47,61);(1,5)(2,11)(3,16)(4,21)(6,26)(7,28)(8,25)(9,30)(10,34)(12,36)(13,24)(14,38)(15,40)(17,45)(18,44)(19,47)(20,49)(22,52)(23,50)(27,51)(29,55)(31,43)(32,53)(33,57)(35,58)(37,54)(39,59)(41,61)(42,60)(46,62)(48,63)(56,64)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303109, #72303351, #72303359
maximal normal subgroup of	#72303109, #72303351, #72303359

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299909);