#364215

order	512
nilpotency class	3
isoclinism	isoclinic class of #363903
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2297
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13961
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

867 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)(3,13,17,36)(4,9)(5,21,23,49)(6,11)(7,12)(8,16,27,44)(10,22,33,50)(14,37,40,34)(15,41)(18,45,47,62)(19,30)(20,31)(24,32)(25,51,28,39)(26,43)(29,46,55,63)(35,57)(38,52)(42,58)(48,54)(53,61)(56,60)(59,64);(1,3)(2,8,12,27)(4,14)(5,15)(6,16)(7,17)(9,28,31,25)(10,26,33,41)(11,13,32,36)(18,38)(19,39)(20,40)(21,35,49,58)(22,42)(23,43)(24,44)(29,53,55,52)(30,37,54,34)(45,59,62,56)(46,60)(47,61)(48,51)(50,57)(63,64);(1,4,6,19,7,20,24,48)(2,9,11,30,12,31,32,54)(3,14,16,39,17,40,44,51)(5,18,22,46,23,47,50,63)(8,25,13,34,27,28,36,37)(10,29,21,45,33,55,49,62)(15,38,42,60,43,61,57,64)(26,52,35,56,41,53,58,59);(1,5,6,22,7,23,24,50)(2,10,32,49,12,33,11,21)(3,15,16,42,17,43,44,57)(4,18,19,46,20,47,48,63)(8,26,36,58,27,41,13,35)(9,29,54,62,31,55,30,45)(14,38,39,60,40,61,51,64)(25,52,37,59,28,53,34,56)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285582, #72285922, #72285996, …
maximal normal subgroup of	#72285582, #72285922, #72285996, …
maximal quotient of	C₂×#364215

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271411);