#129353

order	512
nilpotency class	4
isoclinism	isoclinic class of #12705
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8301
prime radical	C₂×#525	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

848 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36549);