#474442

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16976
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

575 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381638);