#138354

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

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

normal subgroups only

This group has 554 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=gh,e^2=i,f^2=ghi,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, 45550);