#116793

order	512
nilpotency class	3
isoclinism	isoclinic class of #116789
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	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

521 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)(3,13)(4,17,21,27)(5,11)(6,10)(7,12)(8,15)(9,20,28,19)(14,29,31,23)(16,24)(18,32,30,26)(22,25)(33,34,38,42,39,44,37,43)(35,40,47,55,48,56,46,60)(36,49,51,41,52,58,53,59)(45,57,61,54,62,64,50,63);(1,3,5,15,7,16,6,22)(2,8,10,13,12,25,11,24)(4,18,19,14,21,30,20,31)(9,26,17,29,28,32,27,23)(33,35)(34,40)(36,50)(37,46)(38,47)(39,48)(41,57)(42,55)(43,60)(44,56)(45,51)(49,63)(52,61)(53,62)(54,58)(59,64);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,23)(10,17)(11,27)(12,28)(13,26)(15,30)(16,31)(18,22)(24,32)(25,29)(33,36)(34,41)(35,45)(37,53)(38,51)(39,52)(40,54)(42,58)(43,49)(44,59)(46,50)(47,61)(48,62)(55,64)(56,63)(57,60)

Polycyclic presentation:

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

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, 23989);