#498172

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	#22870
prime radical	#33538	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

725 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405368);