#495508

order	512
nilpotency class	3
isoclinism	isoclinic class of #363904
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2297
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21379
prime radical	#3455	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

753 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:

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

Polycyclic presentation:

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

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, 402704);