#151120

order	512
nilpotency class	5
isoclinism	isoclinic class of #151119
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12078
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	#346	C₂³
prime radical	#508	C₂×C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1313 ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

124 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=efhi,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cfg,d^a=df,f^a=fh,h^a=hi,d^b=dfhi,e^b=ehi,f^b=fhi,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225062, #72225113, #72225114, …
maximal normal subgroup of	#72225062, #72225113, #72225114, …
maximal quotient of	C₂×#151120

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58316);