#451647

order	512
nilpotency class	3
isoclinism	isoclinic class of #418871
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2697
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16006
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

739 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)(3,13,17,8)(4,19)(5,25)(6,29)(7,11)(9,23)(10,27)(12,30)(14,42,20,50)(15,45,47,57)(16,33,18,34)(21,53)(22,36)(24,37)(26,39)(28,40)(31,44,41,48)(32,46,38,49)(35,54)(43,60,52,58)(51,55)(56,59)(61,62,64,63);(1,3,6,16,7,17,30,18)(2,8,29,34,11,13,12,33)(4,20,24,44,22,14,23,48)(5,15,26,46,28,47,27,49)(9,31,19,50,37,41,36,42)(10,38,25,57,39,32,40,45)(21,43,56,63,54,52,55,62)(35,58,51,64,53,60,59,61);(1,4,6,24,7,22,30,23)(2,9,29,19,11,37,12,36)(3,14,16,48,17,20,18,44)(5,21,26,56,28,54,27,55)(8,31,34,50,13,41,33,42)(10,35,25,51,39,53,40,59)(15,43,46,63,47,52,49,62)(32,58,45,64,38,60,57,61);(1,5)(2,10)(3,15)(4,21)(6,26)(7,28)(8,32)(9,35)(11,39)(12,40)(13,38)(14,43)(16,46)(17,47)(18,49)(19,51)(20,52)(22,54)(23,55)(24,56)(25,29)(27,30)(31,58)(33,57)(34,45)(36,59)(37,53)(41,60)(42,61)(44,62)(48,63)(50,64)

Polycyclic presentation:

2^9|a^2=efgh,b^2=ef,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^d=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, 358843);