#132097

order	512
nilpotency class	4
isoclinism	isoclinic class of #131979
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#52084	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

626 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39293);