#437442

order	512
nilpotency class	3
isoclinism	isoclinic class of #420065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2686
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

678 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)(3,15)(4,10,24,38)(5,25)(6,13)(7,14)(8,12)(9,18)(11,28)(16,43,45,35)(17,39)(19,33)(20,34)(21,49,48,31)(22,51,54,36)(23,42,30,37)(26,32)(27,40)(29,41)(44,63,52,59)(46,58)(47,55)(50,60,64,57)(53,61,56,62);(1,3,8,18,7,20,6,19)(2,9,12,34,14,33,13,15)(4,21,30,45,24,48,23,16)(5,26,29,47,28,46,27,17)(10,35,37,31,38,43,42,49)(11,39,40,32,25,55,41,58)(22,52,56,64,54,44,53,50)(36,60,61,59,51,57,62,63);(1,4,8,30,7,24,6,23)(2,10,12,37,14,38,13,42)(3,16,18,21,20,45,19,48)(5,22,29,56,28,54,27,53)(9,31,34,43,33,49,15,35)(11,36,40,61,25,51,41,62)(17,44,26,50,47,52,46,64)(32,57,55,63,58,60,39,59);(1,5,7,28)(2,11)(3,17)(4,22,24,54)(6,27,8,29)(9,32,33,58)(10,36)(12,40)(13,41)(14,25)(15,39,34,55)(16,44)(18,26)(19,46)(20,47)(21,50)(23,53,30,56)(31,57,49,60)(35,59,43,63)(37,61)(38,51)(42,62)(45,52)(48,64)

Polycyclic presentation:

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

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, 344638);