#486304

order	512
nilpotency class	4
isoclinism	isoclinic class of #486299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24265	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

916 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393500);