#470189

order	512
nilpotency class	3
isoclinism	isoclinic class of #455198
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2620
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16678
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

608 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:

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

Polycyclic presentation:

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

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, 377385);