#126723

order	512
nilpotency class	4
isoclinism	isoclinic class of #126723
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1736
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#7879
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

331 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,c^b=cgh,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217272, #72217274, #72224958, …
maximal normal subgroup of	#72217272, #72217274, #72224958, …
maximal quotient of	C₂×#126723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33919);