#149516

order	512
nilpotency class	3
isoclinism	isoclinic class of #149515
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1452
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	#346	C₂³
socle	C₂	#11415
prime radical	C₂³	#347
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #346 ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bg,c^a=ci,e^a=eh,g^a=gi,c^b=cf,d^b=dh,f^b=fi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217231, #72225114, #72225116, …
maximal normal subgroup of	#72217231, #72225114, #72225116, …
maximal quotient of	C₂×#149516

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56712);