#388002

order	512
nilpotency class	3
isoclinism	isoclinic class of #388001
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14507
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

813 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 813 conjugacy classes of subgroups — too many to draw. Only its 200 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=eg,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295198);