#137967

order	512
nilpotency class	4
isoclinism	isoclinic class of #137964
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9113
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#32945	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45163);