#116788

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

525 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,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=ei,f^c=fhi

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, 23984);