#368133

order	512
nilpotency class	3
isoclinism	isoclinic class of #23892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14015
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1522 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275329);