#119049

order	512
nilpotency class	3
isoclinism	isoclinic class of #119044
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

482 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)(3,12,15,28,16,8,14,24)(4,17,20,26,21,27,19,9)(13,25,18,23,31,32,30,29)(33,34)(35,45)(36,50,54,58)(37,43)(38,42)(39,44)(40,47)(41,52,59,53)(46,61,63,55)(48,56)(49,60)(51,64,62,57);(1,3,7,16)(2,8,10,12)(4,18,21,30)(5,15,6,14)(9,25,26,32)(11,28,22,24)(13,19,31,20)(17,29,27,23)(33,35,39,49)(34,40,44,56)(36,51,54,62)(37,48,38,47)(41,57,59,64)(42,60,43,45)(46,52,63,53)(50,61,58,55);(1,4)(2,9,10,26)(3,13)(5,19)(6,20)(7,21)(8,23,12,29)(11,17,22,27)(14,30)(15,18)(16,31)(24,32,28,25)(33,36,37,52,39,54,38,53)(34,41,42,58,44,59,43,50)(35,46,47,62,49,63,48,51)(40,55,45,57,56,61,60,64)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,f^c=fhi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26245);