#142304

order	512
nilpotency class	4
isoclinism	isoclinic class of #142304
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9915
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#2190	C₂
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49500);