#494076

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1832
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

533 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265417, #72265430, #72265447
maximal normal subgroup of	#72265417, #72265430, #72265447

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401272);