#129475

order	512
nilpotency class	4
isoclinism	isoclinic class of #129411
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	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1182 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36671);