#497858

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	D₄×#107
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

579 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=fh,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=de,e^b=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405054);