#433465

order	512
nilpotency class	3
isoclinism	isoclinic class of #418818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15480
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1475 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340661);