#118921

order	512
nilpotency class	3
isoclinism	isoclinic class of #118916
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

491 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)(3,11,14,7)(4,8,16,24)(5,10)(6,9)(12,29)(13,27,18,22)(15,32)(17,25,19,28)(20,26)(21,30)(23,31)(33,34,38,44,39,43,37,42)(35,45,48,40,49,57,47,56)(36,50,52,60,53,41,54,59)(46,62,63,55,51,61,64,58);(1,3)(2,7)(4,15)(5,18)(6,14)(8,23)(9,11)(10,27)(12,17)(13,20)(16,31)(19,30)(21,25)(22,26)(24,32)(28,29)(33,35,39,49)(34,40,43,56)(36,51,53,46)(37,48,38,47)(41,58,50,55)(42,57,44,45)(52,63,54,64)(59,61,60,62);(1,4)(2,8,9,24)(3,12)(5,19)(6,16)(7,21,11,29)(10,28,26,25)(13,31)(14,30)(15,18)(17,20)(22,32,27,23)(33,36,37,52,39,53,38,54)(34,41,42,59,43,50,44,60)(35,46,47,63,49,51,48,64)(40,55,45,61,56,58,57,62)

Polycyclic presentation:

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

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, 26117);