#150947

order	512
nilpotency class	5
isoclinism	isoclinic class of #7447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1361
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂	C₂×#1361
prime radical	#33713	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

279 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=ghi,e^2=fgh,f^2=hi,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,h^a=hi,d^b=dg,e^b=efgh,f^b=fh,g^b=ghi,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225338, #72225340, #72225344, …
maximal normal subgroup of	#72225338, #72225340, #72225344, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58143);