#138661

order	512
nilpotency class	4
isoclinism	isoclinic class of #138660
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9221
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#2213	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 378 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214066, #72214153, #72214250
maximal normal subgroup of	#72214066, #72214153, #72214250
maximal quotient of	#72216471, #72227246, #72227247

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45857);