#132098

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 590 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,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, 39294);