#203078

order	512
nilpotency class	3
isoclinism	isoclinic class of #19368
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

597 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110274);