#152307

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	#2163
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

312 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(5,6,8,15,9,16,20,30,10,13,21,26,22,27,33,35,11,12,19,24,23,25,32,34,7,14,17,28,18,29,31,36);(6,12)(8,19)(13,14)(16,25)(17,21)(20,32)(27,29)(31,33);(2,4)(6,13,12,14)(8,19)(15,28,24,26)(16,27,25,29)(17,21)(20,32)(30,36,34,35)(31,33)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59503);