#497895

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	C₄⋊C₄×D₈
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

583 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:

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

Polycyclic presentation:

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

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, 405091);