#151732

order	512
nilpotency class	5
isoclinism	isoclinic class of #151712
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₈

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

166 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)(19,28)(20,29)(21,32)(24,31)(25,40)(26,43)(27,37)(30,39)(33,46)(34,35)(36,41)(38,47)(42,48)(44,45);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9);(17,18,24,38,20,25,30,36,32,41,45,40,46,47,19,23,22,27,28,35,33,42,44,43,21,26,39,48,29,34,31,37)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=egi,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

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216261, #72216265, #72216272, …
maximal normal subgroup of	#72216261, #72216265, #72216272, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58928);