#134059

order	512
nilpotency class	4
isoclinism	isoclinic class of #133825
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1239 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 1239 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,f^c=fh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41255);