#494428

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#386
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1869
prime radical	C₂×#575	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1263 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:

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

Polycyclic presentation:

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

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, 401624);