#138353

order	512
nilpotency class	4
isoclinism	isoclinic class of #138352
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9155
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

546 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45549);