#118885

order	512
nilpotency class	3
isoclinism	isoclinic class of #118880
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

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,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,c^b=cf,d^b=di,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, 26081);