#152790

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

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

206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,24,19,25,32,39,20,15,26,27,33,28,40,41,17,23,30,37,31,38,42,44,16,22,21,34,29,36,35,43);(6,9)(10,12)(13,15,18,27,19,28,32,41,20,23,26,37,33,38,40,44,17,22,30,34,31,36,42,43,16,14,21,24,29,25,35,39);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,22,23,15)(18,21,30,26)(19,29,31,33)(24,34,37,27)(25,36,38,28)(32,35,42,40)(39,43,44,41)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=d,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce

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, 59986);