#148969

order	512
nilpotency class	4
isoclinism	isoclinic class of #140362
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1941
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11257
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

439 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)(3,9,15,35)(4,19)(5,24)(6,26)(7,12)(8,11)(10,21)(13,30)(14,44,45,62)(16,40,18,33)(17,49,47,34)(20,52,54,63)(22,43)(23,39)(25,41)(27,58,48,42)(28,59)(29,57)(31,37)(32,51,61,46)(36,50,64,55)(38,60)(53,56);(1,3)(2,9,11,35)(4,20,23,54)(5,18)(6,27)(7,16)(8,15)(10,36)(12,40,24,33)(13,42,26,58)(14,22,45,56)(17,29)(19,52)(21,55,31,50)(25,47)(28,46,60,51)(30,48)(32,59)(34,57,49,41)(37,64)(38,61)(39,63)(43,62)(44,53);(1,4)(2,10,11,37)(3,14,15,45)(5,21)(6,28)(7,31)(8,23)(9,32)(12,19,24,39)(13,43,26,53)(16,46,18,51)(17,50,47,55)(20,48,54,27)(22,29)(25,56)(30,60)(33,44)(34,52)(35,61)(36,42)(38,57,59,41)(40,62)(49,63)(58,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=gi,h^b=hi,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307185, #72307189, #72307268, …
maximal normal subgroup of	#72307185, #72307189, #72307268, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56165);