#151659

order	512
nilpotency class	5
isoclinism	isoclinic class of #151658
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12447
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#508	D₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

136 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(2,10)(3,12,19,28)(4,7)(5,16)(6,8)(11,20)(13,22)(14,15)(17,29,31,32)(18,30,24,27)(21,23,25,26);(1,2,11,22,6,13,4,10,9,16,7,14,8,15,20,5)(3,12,26,32,18,27,17,25,19,28,23,29,24,30,31,21);(1,3)(2,12)(4,17)(5,21)(6,18)(7,23)(8,24)(9,19)(10,25)(11,26)(13,27)(14,29)(15,30)(16,28)(20,31)(22,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=dfi,d^2=fi,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fi,g^a=gh,h^a=hi,e^c=egi,g^c=gh,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214100, #72214196, #72214234
maximal normal subgroup of	#72214100, #72214196, #72214234
maximal quotient of	#72216267, #72216273

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58855);