#118657

order	512
nilpotency class	3
isoclinism	isoclinic class of #118653
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

515 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)(4,18,22,28)(5,11)(6,10)(7,12)(8,15)(9,21,26,20)(14,23,31,32)(16,24)(17,27)(19,29,30,25)(33,34,38,42,39,43,37,52)(35,40,47,55,46,56,45,61)(36,48,51,41,50,58,53,57)(44,59,60,54,49,63,62,64);(1,3,5,15,7,17,6,16)(2,8,10,13,12,24,11,27)(4,19,20,14,22,30,21,31)(9,25,18,23,26,29,28,32)(33,35)(34,40)(36,49,50,44)(37,45)(38,47)(39,46)(41,54,57,64)(42,55)(43,56)(48,59,58,63)(51,62,53,60)(52,61);(1,4,7,22)(2,9,12,26)(3,14,17,31)(5,20,6,21)(8,23,24,32)(10,18,11,28)(13,29,27,25)(15,30,16,19)(33,36)(34,41)(35,44)(37,53)(38,51)(39,50)(40,54)(42,58)(43,57)(45,62)(46,49)(47,60)(48,52)(55,63)(56,64)(59,61)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=hi,f^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=eh,f^b=fhi,d^c=di,e^c=ehi,f^c=fh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25853);