#114861

order	512
nilpotency class	3
isoclinism	isoclinic class of #114856
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

757 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22057);