#495018

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#121
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1939
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

653 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402214);