#119159

order	512
nilpotency class	3
isoclinism	isoclinic class of #119158
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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,e^c=ehi,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, 26355);