#463156

order	512
nilpotency class	3
isoclinism	isoclinic class of #463155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16465
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 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:

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

Polycyclic presentation:

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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370352);