#389038

order	512
nilpotency class	3
isoclinism	isoclinic class of #389036
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14533
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

1202 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319687, #72319988, #72320118
maximal normal subgroup of	#72319687, #72319988, #72320118

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 296234);