#232124

order	512
nilpotency class	3
isoclinism	isoclinic class of #23360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#518
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

540 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 196 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,5,4,6,7,8)(9,10)(11,19,22,15)(12,23,27,38)(13,17)(14,18)(16,26,32,36)(20,31)(21,33,28,30)(24,29)(25,34)(35,39)(37,40);(9,11,14,22)(10,15,18,19)(12,24,27,39)(13,28,25,21)(16,31,32,40)(17,33,34,30)(20,36,37,26)(23,35,38,29);(9,12,25,36,14,27,13,26)(10,16,17,23,18,32,34,38)(11,20,28,39,22,37,21,24)(15,29,30,40,19,35,33,31);(10,17,18,34)(15,30,19,33)(16,23,32,38)(29,40,35,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,c^b=cef,d^b=de,f^b=fh,g^b=gh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139320);