#116506

order	512
nilpotency class	3
isoclinism	isoclinic class of #116502
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 511 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)(3,12)(4,16,18,26)(5,19)(6,10)(7,11)(8,14)(9,17,27,21)(13,25,31,22)(15,23)(20,24)(28,30,32,29)(33,34,37,42,38,41,50,56)(35,39,45,53,44,52,60,63)(36,46,47,57,48,40,49,55)(43,51,59,62,58,54,61,64);(1,3,5,14,7,15,6,20)(2,8,10,12,11,24,19,23)(4,13,17,29,18,31,21,30)(9,25,26,32,27,22,16,28)(33,35)(34,39)(36,43)(37,45)(38,44)(40,54)(41,52)(42,53)(46,51)(47,59)(48,58)(49,61)(50,60)(55,64)(56,63)(57,62);(1,4)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,26)(11,27)(12,28)(14,29)(15,31)(16,19)(20,30)(23,32)(24,25)(33,36,38,48)(34,40,41,46)(35,43,44,58)(37,49,50,47)(39,51,52,54)(42,57,56,55)(45,61,60,59)(53,64,63,62)

Polycyclic presentation:

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

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, 23702);