#498043

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22799
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405239);