#463154

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16465
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

898 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:

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

Polycyclic presentation:

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

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, 370350);