#151971

order	512
nilpotency class	5
isoclinism	isoclinic class of #13657
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2177
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12637
prime radical	C₂×#508	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

283 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 104 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)(3,15,18,46)(4,11)(5,26)(6,29)(7,32)(8,14)(9,17,34,52)(10,24)(12,31)(13,30)(16,44,21,59)(19,55,53,45)(20,57,54,62)(22,40)(23,41)(25,61)(27,38)(28,39)(33,51,37,48)(35,50,63,58)(36,49,64,56)(42,47)(43,60);(1,3,5,19,27,54,60,48,30,16,47,49,31,58,22,52,8,18,24,53,6,20,25,51,7,21,28,56,23,50,4,17)(2,9,11,35,41,64,39,44,32,33,61,62,29,55,10,15,14,34,40,63,12,36,42,59,13,37,43,57,38,45,26,46);(3,16,18,21)(4,22)(5,24)(9,33,34,37)(10,26)(11,40)(15,44,46,59)(17,51,52,48)(19,56,53,49)(20,50,54,58)(25,60)(28,47)(35,57,63,62)(36,45,64,55)(39,42)(43,61)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=ghi,f^2=gi,g^2=hi,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gh,h^a=hi,c^b=ch,e^b=ei,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223653, #72223679, C₂×#151971, …
maximal normal subgroup of	#72223653, #72223679, C₂×#151971, …
maximal quotient of	#72223651, #72223678, C₂×#151971

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59167);