#184377

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

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

594 conjugacy classes of subgroups; only the 326 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 326 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eg,d^2=ef,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,c^b=chi,d^b=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91573);