#119037

order	512
nilpotency class	3
isoclinism	isoclinic class of #119032
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

485 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:

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

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=fi,c^b=cf,d^b=di,e^b=ei,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, 26233);