#139699

order	512
nilpotency class	4
isoclinism	isoclinic class of #139572
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1910
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₄×M₁₆	C₂³
socle	C₂	#9492
prime radical	#2017	C₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

335 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339870, #72343154
maximal normal subgroup of	#72339870, #72343154

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46895);