#132968

order	512
nilpotency class	4
isoclinism	isoclinic class of #132521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

472 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40164);