#135542

order	512
nilpotency class	4
isoclinism	isoclinic class of #135510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1819
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

588 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,e^c=egi,f^c=fhi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42738);