#134213

order	512
nilpotency class	4
isoclinism	isoclinic class of #133922
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2315 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

924 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,f^b=fg,e^c=ei,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, 41409);