#119065

order	512
nilpotency class	3
isoclinism	isoclinic class of #119062
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

497 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)(3,11,13,7)(4,15)(5,19)(6,9)(8,17)(10,22)(12,29,30,23)(14,24,20,27)(16,25,31,32)(18,26)(21,28)(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)(2,7)(4,16)(5,20)(6,13)(8,25)(9,11)(10,27)(12,18)(14,22)(15,32)(17,31)(19,24)(21,30)(23,26)(28,29)(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)(2,8,9,15)(3,12)(5,21)(6,17)(7,23,11,29)(10,28,19,26)(13,30)(14,31)(16,20)(18,22)(24,32,27,25)(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,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,e^c=eh,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, 26261);