#118497

order	512
nilpotency class	3
isoclinism	isoclinic class of #118493
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	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

516 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)(5,11)(6,10)(7,12)(8,15)(9,18)(14,22)(16,26)(19,27)(20,23)(21,25)(24,31)(28,30)(29,32)(33,34,38,44,39,43,37,42)(35,40,48,56,47,57,46,55)(36,49,53,41,51,59,52,58)(45,60,62,54,50,63,61,64);(1,3,5,15,7,16,6,20)(2,8,10,13,12,23,11,26)(4,14,18,29,19,31,21,30)(9,24,17,32,25,22,27,28)(33,35)(34,40)(36,50,51,45)(37,46)(38,48)(39,47)(41,54,58,64)(42,55)(43,57)(44,56)(49,60,59,63)(52,62,53,61);(1,4,7,19)(2,9,12,25)(3,14,16,31)(5,18,6,21)(8,22,23,24)(10,17,11,27)(13,28,26,32)(15,29,20,30)(33,36)(34,41)(35,45)(37,53)(38,52)(39,51)(40,54)(42,59)(43,58)(44,49)(46,62)(47,50)(48,61)(55,63)(56,60)(57,64)

Polycyclic presentation:

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

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, 25693);