#471070

order	512
nilpotency class	3
isoclinism	isoclinic class of #471069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16741
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

702 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)(4,21)(5,27,31,42)(6,13)(7,12)(8,14)(9,18)(10,24)(11,30,41,29)(16,45)(17,33,52,59)(19,35)(20,34)(22,36)(23,53,55,37)(25,39)(26,38)(28,40,54,60)(32,48)(43,47)(44,50,56,51)(46,58,64,61)(49,57,63,62);(1,3,7,18,8,20,6,19)(2,9,12,34,14,35,13,15)(4,22,25,48,26,47,24,16)(5,17,30,50,31,52,29,51)(10,36,21,32,39,43,38,45)(11,33,27,56,41,59,42,44)(23,49,54,64,55,63,28,46)(37,58,40,57,53,61,60,62);(1,4,8,26)(2,10,14,39)(3,16,20,48)(5,28,31,54)(6,25,7,24)(9,32,35,45)(11,40,41,60)(12,38,13,21)(15,43,34,36)(17,49,52,63)(18,47,19,22)(23,30,55,29)(27,37,42,53)(33,58,59,61)(44,57,56,62)(46,51,64,50);(1,5,6,29,8,31,7,30)(2,11,12,27,14,41,13,42)(3,17,19,51,20,52,18,50)(4,23,24,28,26,55,25,54)(9,33,34,56,35,59,15,44)(10,37,21,40,39,53,38,60)(16,46,47,63,48,64,22,49)(32,57,43,61,45,62,36,58)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285951, #72286106, #72286154
maximal normal subgroup of	#72285951, #72286106, #72286154

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378266);