#133048

order	512
nilpotency class	4
isoclinism	isoclinic class of #132549
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×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 506 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,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, 40244);