#113143

order	512
nilpotency class	3
isoclinism	isoclinic class of #10496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

757 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh,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, 20339);