#498292

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	D₄²
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22985
prime radical	#3067	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

604 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225458, #72225467, #72225469
maximal normal subgroup of	#72225458, #72225467, #72225469

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405488);