#354033

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂³	C₂²×C₄²
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2938 conjugacy classes of subgroups; only the 738 normal subgroups are shown.

normal subgroups only

This group has 2938 conjugacy classes of subgroups — too many to draw. Only its 738 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fgi,d^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261229);