#133253

order	512
nilpotency class	4
isoclinism	isoclinic class of #133175
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 442 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=gi,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,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, 40449);