#382345

order	512
nilpotency class	3
isoclinism	isoclinic class of #30236
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2644
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14331
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 688 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=fgh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=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, 289541);