#364559

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13967
prime radical	#60398	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2351 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317339, #72317364
maximal normal subgroup of	#72317339, #72317364

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271755);