#116926

order	512
nilpotency class	3
isoclinism	isoclinic class of #116918
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

531 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24122);