#189008

order	512
nilpotency class	3
isoclinism	isoclinic class of #188968
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

751 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)(4,17,19,24)(5,20)(6,10)(7,11)(8,14)(9,18,25,21)(13,22,29,32)(15,23)(16,27)(26,28,31,30)(33,34,37,42,38,41,50,55)(35,39,45,53,44,52,60,63)(36,46,47,56,48,40,49,54)(43,57,59,64,58,51,61,62);(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)(36,43)(37,45)(38,44)(40,51)(41,52)(42,53)(46,57)(47,59)(48,58)(49,61)(50,60)(54,62)(55,63)(56,64);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,25)(12,26)(14,28)(15,30)(16,29)(17,20)(23,32)(27,31)(33,36,38,48)(34,40,41,46)(35,43,44,58)(37,49,50,47)(39,51,52,57)(42,56,55,54)(45,61,60,59)(53,64,63,62);(35,44)(39,52)(43,58)(45,60)(51,57)(53,63)(59,61)(62,64)

Polycyclic presentation:

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

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, 96204);