#129304

order	512
nilpotency class	4
isoclinism	isoclinic class of #12725
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8300
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

660 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,f^b=fg,e^c=ei,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, 36500);