#135482

order	512
nilpotency class	4
isoclinism	isoclinic class of #135477
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1819
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 531 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42678);