#119324

order	512
nilpotency class	3
isoclinism	isoclinic class of #119322
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

500 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26520);