#489387

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#62436	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

781 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 781 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

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, 396583);