#193171

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

753 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,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, 100367);