#402480

order	512
nilpotency class	3
isoclinism	isoclinic class of #394482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2435
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14843
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1062 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309676);