#119344

order	512
nilpotency class	3
isoclinism	isoclinic class of #119339
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

502 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=ghi,e^c=eh,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, 26540);