#213893

order	512
nilpotency class	3
isoclinism	isoclinic class of #213828
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1340 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,g^c=gi,g^d=gh

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, 121089);