#132180

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	#52059	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#542 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 628 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,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, 39376);