#119327

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	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

517 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)(35,45)(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);(1,3,7,17)(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)(33,35,39,49)(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)(3,14)(5,19)(6,21)(7,20)(8,22,13,25)(11,18,12,27)(15,29)(16,31)(17,30)(23,32,24,28)(33,36,37,52,39,54,38,53)(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=i,g^2=hi,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, 26523);