#483513

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂²×D₈
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂	C₂×C₈×D₈
prime radical	C₂×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

475 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320266, #72320271, #72320299, …
maximal normal subgroup of	#72320266, #72320271, #72320299, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390709);