#133056

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	C₂×#117	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 491 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,e^2=i,f^2=hi,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, 40252);