#433978

order	512
nilpotency class	3
isoclinism	isoclinic class of #418818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15494
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

975 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)(5,25,28,11)(6,13)(7,14)(8,12)(9,18)(16,44,46,35)(17,39,47,43)(19,33)(20,34)(21,50,49,31)(22,36)(23,42,30,37)(26,32,48,58)(27,40,29,41)(45,62)(51,64)(52,61)(53,60)(54,55)(56,57)(59,63);(1,3)(2,9)(4,21)(5,26)(6,19)(7,20)(8,18)(10,35)(11,39)(12,34)(13,15)(14,33)(16,23)(17,27)(22,45)(24,49)(25,43)(28,48)(29,47)(30,46)(31,37)(32,40)(36,57)(38,44)(41,58)(42,50)(51,52)(53,56)(54,63)(55,64)(59,60)(61,62);(1,4,8,30,7,24,6,23)(2,10,12,37,14,38,13,42)(3,16,18,21,20,46,19,49)(5,22,27,52,28,54,29,53)(9,31,34,44,33,50,15,35)(11,36,41,61,25,55,40,60)(17,45,48,51,47,63,26,56)(32,57,39,62,58,64,43,59);(1,5)(2,11)(3,17)(4,22)(6,29)(7,28)(8,27)(9,32)(10,36)(12,41)(13,40)(14,25)(15,43)(16,45)(18,48)(19,26)(20,47)(21,51)(23,53)(24,54)(30,52)(31,57)(33,58)(34,39)(35,59)(37,61)(38,55)(42,60)(44,62)(46,63)(49,56)(50,64)

Polycyclic presentation:

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

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, 341174);