#118256

order	512
nilpotency class	3
isoclinism	isoclinic class of #10560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

520 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)(3,12)(4,17,19,24)(5,20)(6,10)(7,11)(8,14)(9,18,25,21)(13,22,29,32)(15,23)(16,27)(26,28,31,30)(33,34,37,42,38,41,49,55)(35,39,45,51,44,52,57,62)(36,40,48,54,47,53,60,63)(43,50,56,61,46,58,59,64);(1,3,5,14,7,16,6,15)(2,8,10,12,11,23,20,27)(4,13,18,28,19,29,21,30)(9,22,24,26,25,32,17,31)(33,35)(34,39)(36,46,47,43)(37,45)(38,44)(40,50,53,58)(41,52)(42,51)(48,59,60,56)(49,57)(54,61,63,64)(55,62);(1,4,7,19)(2,9,11,25)(3,13,16,29)(5,18,6,21)(8,22,23,32)(10,24,20,17)(12,26,27,31)(14,28,15,30)(33,36)(34,40)(35,43)(37,48)(38,47)(39,50)(41,53)(42,54)(44,46)(45,56)(49,60)(51,61)(52,58)(55,63)(57,59)(62,64)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,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, 25452);