#119854

order	512
nilpotency class	3
isoclinism	isoclinic class of #119849
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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27050);