#116800

order	512
nilpotency class	3
isoclinism	isoclinic class of #116795
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

524 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 23996);