#153580

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	C₂³	C₂×C₃₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,11,17,12,18,21,35,13,19,22,36,25,38,41,57,14,20,23,37,26,39,42,58,28,40,44,59,47,60,61,70,15,16,24,31,27,32,43,51,29,33,45,52,48,54,62,66,30,34,46,53,49,55,63,67,50,56,64,68,65,69,71,72);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,16)(17,31)(18,32)(19,33)(20,34)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(57,66)(58,67)(59,68)(60,69)(70,72)

Polycyclic presentation:

2^9|a^2=e,c^2=d,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=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, 60776);