#498314

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
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	#22995
prime radical	#33594	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

607 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)(3,16,20,43)(4,22)(5,11)(6,12)(7,31)(8,15)(9,19,37,21)(10,25)(13,29)(14,30)(17,47,51,62)(18,41,52,36)(23,40)(24,44)(26,38)(27,56)(28,59,53,35)(32,45)(33,42)(34,48,60,64)(39,58)(46,57,61,50)(49,63,54,55);(1,3,6,21,8,20,30,19)(2,9,12,43,15,37,14,16)(4,17,24,54,27,51,26,49)(5,28,7,18,29,53,33,52)(10,34,39,61,40,60,45,46)(11,41,31,35,13,36,42,59)(22,55,44,62,56,63,38,47)(23,57,32,48,25,50,58,64);(1,4)(2,10)(3,17)(5,23)(6,24)(7,32)(8,27)(9,34)(11,38)(12,39)(13,44)(14,45)(15,40)(16,46)(18,48)(19,49)(20,51)(21,54)(22,31)(25,29)(26,30)(28,57)(33,58)(35,55)(36,62)(37,60)(41,47)(42,56)(43,61)(50,53)(52,64)(59,63);(1,5,6,7,8,29,30,33)(2,11,12,31,15,13,14,42)(3,18,19,28,20,52,21,53)(4,23,24,32,27,25,26,58)(9,35,16,41,37,59,43,36)(10,38,39,22,40,44,45,56)(17,48,49,57,51,64,54,50)(34,55,46,47,60,63,61,62)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225388, #72225396, #72225403
maximal normal subgroup of	#72225388, #72225396, #72225403

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405510);