#371030

order	512
nilpotency class	3
isoclinism	isoclinic class of #23996
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2404
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14064
prime radical	#3449	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1239 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317342, #72317348, #72317353
maximal normal subgroup of	#72317342, #72317348, #72317353

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278226);