#391508

order	512
nilpotency class	3
isoclinism	isoclinic class of #32873
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2840
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14635
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 826 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,f^c=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, 298704);