#150147

order	512
nilpotency class	5
isoclinism	isoclinic class of #150146
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11663
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	#346	C₂³
prime radical	#3414	C₄
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

226 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,14,15,45,18,47,19,43)(4,9)(5,23,27,57)(6,29)(7,13)(8,16,32,50,34,53,31,48)(10,28,40,61)(11,21)(12,22)(17,37,30,62,51,64,55,46)(20,56,25,42)(24,41,58,44,59,63,52,33)(26,36)(35,49,38,60)(39,54);(1,3)(2,8)(4,15)(5,24)(6,30)(7,18)(9,31)(10,37)(11,41)(12,32)(13,34)(14,42)(16,49)(17,54)(19,22)(20,52)(21,51)(23,45)(25,58)(26,55)(27,59)(28,50)(29,44)(33,36)(35,62)(38,46)(39,63)(40,64)(43,57)(47,56)(48,61)(53,60);(1,4,7,22)(2,9,13,12)(3,15,18,19)(5,25,27,20)(6,21,26,54)(8,31,34,32)(10,38,40,35)(11,36,39,29)(14,43,47,45)(16,50,53,48)(17,30,51,55)(23,42,57,56)(24,58,59,52)(28,60,61,49)(33,63,44,41)(37,46,64,62)

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=fi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,d^b=dfh,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225029, #72225113, #72225116, …
maximal normal subgroup of	#72225029, #72225113, #72225116, …
maximal quotient of	C₂×#150147

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57343);