#151733

order	512
nilpotency class	5
isoclinism	isoclinic class of #151715
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,8)(3,9,12,16)(4,13,10,7)(6,14,15,11)(18,23)(19,25)(20,28)(21,31)(24,33)(26,42)(27,43)(29,45)(30,37)(32,44)(34,35)(36,39)(38,48)(40,41)(46,47);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(18,24)(23,33)(26,38)(27,39)(34,46)(35,47)(36,43)(42,48);(17,18,25,35,20,26,40,43,31,39,45,48,44,46,30,33,22,24,37,47,32,38,29,36,21,27,41,42,28,34,19,23)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eg,d^2=f,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh,h^a=hi,c^b=ci

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58929);