#133219

order	512
nilpotency class	4
isoclinism	isoclinic class of #133089
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 638 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)(3,13,17,34)(4,18,20,9)(5,22,25,10)(6,11)(7,12)(8,15,29,47)(14,41)(16,40,48,39)(19,52,54,32)(21,51,56,33)(23,30,46,31)(24,50,55,35)(26,57,59,36)(27,37)(28,42,62,45)(38,64)(43,63)(44,60)(49,53,61,58);(1,3,6,17)(2,8,11,31)(4,14,20,44)(5,23,26,15)(7,16,27,48)(9,28,18,49)(10,34,36,13)(12,30,37,29)(19,53,55,42)(21,43,56,64)(22,39,57,40)(24,58,54,45)(25,47,59,46)(32,41,50,60)(33,62,51,61)(35,63,52,38);(1,4,7,21)(2,9,12,33)(3,14,16,43)(5,24,25,19)(6,20,27,56)(8,28,30,62)(10,35,22,32)(11,18,37,51)(13,38,40,60)(15,45,46,42)(17,44,48,64)(23,58,47,53)(26,54,59,55)(29,61,31,49)(34,63,39,41)(36,52,57,50)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40415);