#153585

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₃₂	C₂³
commutator, socle	C₂²	C₂²×C₃₂
prime radical	C₂×D₄	C₃₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,9,15,10,16,27,39,11,17,28,40,29,41,53,62,12,7,18,20,30,21,42,46,31,22,43,47,54,48,63,65,8,13,23,32,24,33,49,55,25,34,50,56,51,57,66,67,26,19,14,35,36,44,37,58,52,45,38,59,60,64,61,68);(2,4)(6,13)(7,19)(15,32)(16,33)(17,34)(20,35)(21,44)(22,45)(39,55)(40,56)(41,57)(46,58)(47,59)(48,64)(62,67)(65,68);(5,7,9,20,10,21,27,46,11,22,28,47,29,48,53,65,12,13,18,32,30,33,42,55,31,34,43,56,54,57,63,67,8,19,23,35,24,44,49,58,25,45,50,59,51,64,66,68,26,6,14,15,36,16,37,39,52,17,38,40,60,41,61,62)

Polycyclic presentation:

2^9|a^2=e,c^2=e,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,c^b=ci

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60781);