#151060

order	512
nilpotency class	5
isoclinism	isoclinic class of #151059
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12030
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	#346	C₂³
prime radical	C₂³	#365
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

118 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=i,d^2=f,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cgh,e^b=ehi,g^b=gi,d^c=di,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224979, #72225113, #72225116, …
maximal normal subgroup of	#72224979, #72225113, #72225116, …
maximal quotient of	C₂×#151060

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58256);