#111160

order	512
nilpotency class	3
isoclinism	isoclinic class of #111151
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

565 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)(5,21)(6,10)(7,11)(8,14)(9,18)(13,22)(15,24)(16,23)(19,25)(20,31)(26,28)(27,30)(29,32)(33,34)(35,39,44,54)(36,46)(37,50)(38,41)(40,49)(42,47)(43,57,59,60)(45,56,51,53)(48,61)(52,63,64,58)(55,62);(1,3,6,14,7,16,5,15)(2,8,10,23,11,24,21,12)(4,13,19,27,20,29,18,28)(9,22,17,30,25,32,31,26)(33,35)(34,39,41,54)(36,43)(37,51)(38,44)(40,52,55,64)(42,56,50,53)(45,47)(46,60,61,57)(48,59)(49,63)(58,62);(1,4,5,18,7,20,6,19)(2,9,10,17,11,25,21,31)(3,13,15,28,16,29,14,27)(8,22,23,30,24,32,12,26)(33,36,47,62,38,48,37,49)(34,40,50,61,41,55,42,46)(35,43,45,58,44,59,51,63)(39,52,53,57,54,64,56,60)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18356);