#119041

order	512
nilpotency class	3
isoclinism	isoclinic class of #119038
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

495 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)(3,12,16,8)(4,17,20,27)(5,11)(6,21)(7,10)(9,19,26,22)(13,25)(14,28,15,24)(18,23)(29,31)(30,32)(33,34,38,44,39,43,37,42)(35,45,49,40,48,57,47,56)(36,50,52,59,53,41,54,58)(46,60,61,63,51,64,62,55);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,25)(10,12)(11,28)(13,19)(17,29)(20,31)(21,24)(22,30)(23,27)(26,32)(33,35)(34,40)(36,51)(37,49)(38,47)(39,48)(41,55)(42,57)(43,56)(44,45)(46,53)(50,63)(52,61)(54,62)(58,64)(59,60);(1,4)(2,9,10,26)(3,13)(5,19)(6,22)(7,20)(8,23,12,29)(11,17,21,27)(14,18)(15,31)(16,30)(24,32,28,25)(33,36,37,52,39,53,38,54)(34,41,42,58,43,50,44,59)(35,46,47,61,48,51,49,62)(40,55,45,60,56,63,57,64)

Polycyclic presentation:

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

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, 26237);