#134150

order	512
nilpotency class	4
isoclinism	isoclinic class of #133893
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1178 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 1178 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=ei,f^c=fh,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, 41346);